http://www.map.mpim-bonn.mpg.de/api.php?action=feedcontributions&user=Pawel+Walczak&feedformat=atomManifold Atlas - User contributions [en]2022-10-07T12:32:27ZUser contributionsMediaWiki 1.18.4http://www.map.mpim-bonn.mpg.de/Symplectic_manifoldsSymplectic manifolds2010-11-27T16:10:05Z<p>Pawel Walczak: </p>
<hr />
<div>{{Stub}}== Introduction ==<br />
<wikitex>;<br />
A '''symplectic manifold''' is a smooth manifold $M$ together with a differential two-form $\omega$ that is nondegenerate and closed. The form $\omega$ is called a '''symplectic form'''. The nondegeneracy means that the highest nonzero power of $\omega$ is a volume form on $M.$ It follows that a symplectic manifold is even dimensional.<br />
<br />
Symplectic manifolds originated from classical mechanics. The phase space of a dynamical system is the cotangent bundle of the configuration space and it is equipped with a symplectic form. This symplectic form is preserved by the flow of the system.<br />
</wikitex><br />
== Examples ==<br />
<wikitex><br />
$\bullet$<br />
The most basic example of a symplectic manifold is $\mathbb R^{2n}$ equipped with the form <br />
$\omega_0:=dx^1\wedge dy^1 + \ldots + dx^n\wedge dy^n.$ <br />
<br />
A theorem of Darboux {{cite|McDuff&Salamon1998}} states that locally every <br />
symplectic manifold if of this form. More precisely, if $(M,\omega)$ is a symplectic $2n$-manifold<br />
then for every point $x\in M$ there exists an open neighbourhood $U\subset M$ of $p$ and a <br />
diffeomorphism $f\colon U\to f(U)\subset \mathbb R^{2n}$ such that the restriction of $\omega$<br />
to $U$ is equal to the pull-back $f^*\omega_0.$ This implies that symplectic manifolds have<br />
no local invariants.<br />
<br />
$\bullet$<br />
An area form on an oriented surface is symplectic.<br />
<br />
$\bullet$<br />
Let $X$ be a smooth manifold and let $\lambda$ be a one-form on the cotangent bundle $T^*X$ defined as follows.<br />
If $V$ is a vector tangent to $T^*X$ at a point $\alpha$ then $\lambda_{\alpha}(X) = \alpha (\pi_*(X)),$ where<br />
$\pi\colon T^*X\to X$ is the projection. In local coordinates the form $\lambda$ can be expressed as<br />
$\sum y^idx^i.$ The differential $d\lambda$ is a symplectic form on the cotangent bundle $T^*X.$<br />
<br />
$\bullet$<br />
If $(M,\omega)$ is a closed, i.e. compact and without boundary, symplectic $2n$-manifold then the cohomology classes<br />
$[\omega]^k$ are non-zero for $k=0,1.\ldots,n.$ This follows from the fact that the cohomology class of the volume<br />
form $\omega^n$ is nonzero on a closed manifold. This necessary condition implies that spheres of dimension greater than<br />
two are not symplectic. More generally, no closed manifold of the form $M \times S^k$ is symplectic for $k>2.$<br />
<br />
$\bullet$<br />
The complex projective space $\mathbb C \mathbb P^n$ is symplectic with respect to its K\"ahler form.<br />
Its pull back to a complex projective smooth manifold $X \subset \mathbb C \mathbb P^n$ is also symplectic.<br />
More generally, every K\"ahler manifold is symplectic.<br />
</wikitex><br />
<br />
== Symmetries ==<br />
<wikitex>;<br />
A diffeomorphism $f\colon M\to M$ of a symplectic manifold $(M,\omega)$ is called symplectic if it preserves<br />
the symplectic form, $f^*\omega = \omega.$ Sometimes such a diffeomorphism is called a symplectiomorphism.<br />
The group of all symplectic diffeomorphisms of $(M,\omega)$ is denoted by <br />
$\operatorname{Symp}(M,\omega).$<br />
<br />
It follows from the nondegeneracy of the symplectic form $\omega$ the map $X \mapsto \iota_X\omega$ defines an isomorphism<br />
between the vector fields and the one-forms on a symplectic manifold $(M,\omega).$ If the flow of a vector field $X$ <br />
preserves the symplectic form we have that $0 = L_X\omega = d\iota_X \omega + \iota _X d\omega.$ Then the closedness<br />
of the symplectic form implies that the one-form $\iota_X\omega$ is closed. It follows that the Lie algebra of<br />
the group of symplectic diffeomorhism consists of the vector fields $X$ for which the one-form $\iota _X \omega$<br />
is closed. Hence it can be identified with the space of closed one-forms.<br />
<br />
If the one-form $\iota _X \omega$ is exact, i.e. $\iota _X \omega = dH$ for some function $H\colon M\to \mathbb R$<br />
then the vector field $X$ is called Hamiltonian. Symplectic diffeomorphism generated by Hamiltonian flows form<br />
a group $\operatorname{Ham}(M,\omega)$ called the group of Hamiltonian diffeomorphism. Its Lie algebra can be<br />
identified with the quotient of the space of smooth functions on $M$ by the constants.<br />
</wikitex><br />
==Constructions==<br />
===Products=== <br />
<wikitex>;<br />
The product of symplectic manifolds $(M_1,\omega_1)$ and $(M_2,\omega_2)$ is a symplectic manifold with <br />
respect to the form $a\cdot p_1^*\omega_1 + b\cdot p_2^*\omega_2$ for nonzero real numbers <br />
$a,b\in \mathbb R.$ Here $p_i\colon M_1\times M_2\to M_i$ is the projection.<br />
</wikitex><br />
<br />
===Bundles=== <br />
<wikitex>;<br />
A locally trivial bundle $M\to E\to B$ is called symplectic (resp. Hamiltonian) if its structure<br />
group is a subgroup of the group of symplectic (resp. Hamiltonian) diffeomorphisms.<br />
<br />
'''Example.''' The product of the Hopf bundle with the circle is a symplectic bundle $T^2 \to S^3 \times S^1 \to S^2.$<br />
Indeed, the structure group is a group of rotations of the torus and hence it preserves the area. <br />
As we have seen above the product $S^3 \times S^1$ does not admit a symplectic form. This example<br />
shows that, in general, the total space of a symplectic bundle is not symplectic.<br />
<br />
Let $M\stackrel {i}\to E\stackrel{\pi}\to B$ is a compact symplectic bundle over a symplectic base.<br />
According to a theorem of Thurston, if there exists a cohomology class $a\in H^2(E)$ such that<br />
its pull back to every fibre is equal to the class of the symplectic form of the fibre<br />
then there exists a representative $\alpha $ of the class $a$ such that<br />
$\Omega := \alpha + k\cdot \pi^*(\omega_B)$ is a symplectic form on $E$ for every big enough $k.$<br />
<br />
A symplectic fiber bundle may have a symplectic form on the total space which restricts symplectically to the fibers, even if the base is not symplectic. Such bundles are constructed using '''fat connections'''. Let there be given a principal fiber bundle<br />
$$G\rightarrow P\rightarrow M.$$<br />
Let $\theta$ be a connection form, $\Omega$ the curvature form of this connection, and $\mathcal{H}$ be the horizontal distribution. A vector $v\in\frak{g}^*$ is called ''fat'' (with respect to the given connection), if the 2-form <br />
$$(X,Y)\rightarrow \langle \Omega(X,Y),v\rangle$$<br />
is nondegenerate on the horizontal distribution of the connection. The following theorem yields a construction of a symplectic form on the total space of fiber bundles associated with principal bundles equipped with particular connections. <br />
<br />
{{beginthm|Theorem|}} Let there be given a symplectic manifold $(F,\omega)$ endowed with a hamiltonian action of a Lie group $G$. Let $\mu: F\rightarrow\frak{g}^*$ be the moment map of the $G$-action. If $\mu(F)\subset\frak{g}^*$ consists of fat vectors, then the associated bundle <br />
$$F\rightarrow P\times_GF\rightarrow M$$<br />
admits a fiberwise symplectic form on the total space.<br />
{{endthm}}<br />
using this theorem, one can construct examples of symplectic fiber bundles with fiberwise symplectic form on the total space (see examples below).<br />
<br />
'''Example (twistor bundles)'''<br />
Consider the principal bundle of the orthogonal frame bundles over $2n$-dimensional manifold $M$:<br />
$$SO(2n)\rightarrow P\rightarrow M.$$<br />
Let $F=SO(2n)/U(n)$. The associated bundle with fiber $SO(2n)/U(n)$ is called the '''twistor bundle'''. It is easy to see that $SO(2n)/U(n)$ can be identified with a coadjoint orbit $F_{\xi}$ of some $\xi\in\frak{g}^*$, where $\frak{g}^*$ denotes the Lie algebra of $SO(2n)$. Moreover, if $M$ admits Riemannian metric of pinched curvature with sufficiently small pinching constant then $\xi$ is fat with respect to the Levi-Civitta connection in the frame bundle. As a result, the whole coadjoint orbit (which is the image of the moment map of the $SO(2n)$-action) consists of fat vectors. Thus, we obtain a fiberwise symplectic structure on the total space of any twistor bundle <br />
$$F_{\xi}\rightarrow P\times_{SO(2n)}F_{\xi}\rightarrow M$$<br />
over even-dimensional manifolds of pinched curvature. In particular, twistor bundles over spheres $S^{2n}$ or hyperbolic manifolds, admit fiberwise symplectic structures. The simplest example of this construction is the fibering of $\mathbb{C}P^3$ over $S^4$ with fiber $\mathbb{C}P^1$, since it is known that the total space of the twistor bundle over $S^4$ is $\mathbb{C}P^3$. <br />
<br />
'''Example (locally homogeneous complex manifolds)'''<br />
Let $G$ be a Lie group of non-compact type, which is a real form of a complex Lie group $G^c$. Choose a parabolic subgroup $B\subset G^c$ and a maximal compact subgroup $K$ in $G$. Assume that $V=B\subset G$ is compact. Then one can show that $K/V$ can be identified with a coadjoint orbit of some vector in $\frak{k}^*$, which is fat with respect to a $K$-invariant connection in the principal bundle<br />
$$K\rightarrow G/V\rightarrow G/K.$$<br />
It follows that the associated bundle<br />
$$K/V\rightarrow G/V\rightarrow G/K$$<br />
is a symplectic fiber bundle with fiberwise symplectic structure. This construction can be compactified by taking lattices in $G$ which intersect trivially with $K$. A particular example is given by the fiber bundle<br />
$$SO(2n)/U(n)\rightarrow SO(2n,p)/U(n)\times SO(p)\rightarrow SO(2n,p)/SO(2n)\times SO(p),$$<br />
and its compactification by lattices.</wikitex><br />
<br />
===Symplectic reduction===<br />
<wikitex>;<br />
Let $G$ be a Lie group acting on a symplectic manifold $(M,\omega)$ in a hamiltonian way. Denote <br />
by $\mu: M\rightarrow\frak{g}^*$ the moment map of this action. Since $G$ acts on the level set <br />
$\mu^{-1}(a),a\in\frak{g}^*$, one can consider the orbit space $\mu^{-1}(a)/G$. It is an orbifold in general, but it happens to be a manifold, when $G$ acts freely on the preimage, and $a$ is a regular point. In this case, $\tilde M=\mu^{-1}(a)/G$ is a symplectic manifold as well, called '''symplectic reduction'''. It is often denoted by $M//G$.<br />
</wikitex><br />
===Symplectic cut===<br />
<wikitex>;<br />
Let $(M,\omega)$ be a symplectic manifold with a hamiltonian action of the circle $S^1.$ If $\mu:\rightarrow \mathbb R$ is the moment map, $M_a=\mu^{-1}(a)$ is a regular level, then the action restricted to $M_a$ has no fixed points, hence $M_a$ is the boundary of the associated disk bundle W. This is a manifold if the action is free and an orbifold if a non-trivial isotropy occurs.<br />
</wikitex><br />
===Coadjoint orbits===<br />
<br />
===Symplectic homogeneous spaces===<br />
<br />
Nilmanifolds, solvmanifolds, homogeneous spaces of semisimple Lie groups<br />
<br />
===Donaldson's theorem on submanifolds===<br />
<br />
===Surgery in codimension 2===<br />
<wikitex>;<br />
Consider two symplectic manifolds $(M_1,\omega_1),(M_2,\omega_2)$ of equal dimension and suppose that there are codimension two symplectic submanifolds $V_1\subset M_1,V_2\subset M_2$ and a symplectomorphism $f:V_1\rightarrow V_2$ such that Chern classes of normal bundles satisfy $f^*c_1(\nu_2)=-c_1(\nu_1).$ <br />
Then by removing tubular neighborhods of $V_1$ and $V_2$ we get manifolds with boundaries. The map $f$ induces a diffeomorphism of the boundaries, one can form a new manifold identifying the boundaries by this diffeomorphism and define on it a symplectic form which coincides with <br />
$\omega_1$ and $\omega_2$ outside of a tubular neigborhood of the trace of glueing {{cite|Gompf1995}}. The same works if $V_1,V_2$ are symplectic submanifolds of a connected symplectic manifold. </wikitex><br />
<br />
===Symplectic blow-up===<br />
<br />
== Invariants ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Classification/Characterization ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== Further discussion ==<br />
<wikitex>;<br />
...<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
<!-- Please modify these headings or choose other headings according to your needs. --><br />
<br />
[[Category:Manifolds]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T15:02:59Z<p>Pawel Walczak: /* Growth in groups */</p>
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To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
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{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
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END OF COMMENT<br />
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{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have $gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} Any finitely generated abelian group of rank $d$ has the growth type $[n^d]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem}} Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth \cite{Wolf1968}. Conversely, any finitely generated group of polynomial type of growth contains a nilpotent subgroup of finite index \cite{Gromov1981}.<br />
{{endthm}}<br />
<br />
Natural examples of finitely generated groups are provided by<br />
fundamental groups of compact manifolds. Below we show how the growth type<br />
of the fundamental group of a compact Riemannian manifold is related to<br />
its geometry. To this end let us consider a<br />
complete Riemannian manifold $M$, fix a point $x\in M$ and let<br />
$$t_M(n) = vol B(x,n),$$<br />
where $B(x,r)$ is the ball on $M$ of radius $r$ and centre $x$ while $vol$<br />
is the measure on $M$ induced by the Riemannian structure. If $x'$ is another point of $M$, then $B(x',r)\subset B(x, r + r_0)$, where $r_0 =<br />
d(x, x')$ and $d$ is the distance function on $M$.<br />
Therefore, the following definition is correct.<br />
<br />
{{beginthm|Definition}} $gr (M) = [t_M]$ is called the '''growth type''' of a Riemannian manifold $M$. <br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}}\ (\cite{Milnor1968a}, \cite{vSvarc1955}) If $M$ is compact, then the fundamental group $\pi_1 (M)$ and the universal covering $\tilde M$ of $M$ have the same growth type. If $M$ is a compact Riemannian manifold of negative sectional<br />
curvature, then both $\tilde M$ and $\pi_1(M)$ have exponential growth.<br />
{{endthm}}<br />
<br />
Fundamental groups of compact manifolds of negative sectional curvature are hyperbolic in the sense of \cite{Gromov1987}. The last statement of the Proposition above can be generalized as follows,<br />
<br />
{{beginthm|Proposition}} Any non-elementary hyperbolic group has exponential growth. <br />
{{endthm}} <br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T14:57:57Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have $gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} Any finitely generated abelian group of rank $d$ has the growth type $[n^d]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem}} Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth \cite{Wolf1968}. Conversely, any finitely generated group of polynomial type of growth contains a nilpotent subgroup of finite index \cite{Gromov1981}.<br />
{{endthm}}<br />
<br />
Natural examples of finitely generated groups are provided by<br />
fundamental groups of compact manifolds. Below we show how the growth type<br />
of the fundamental group of a compact Riemannian manifold is related to<br />
its geometry. To this end let us consider a<br />
complete Riemannian manifold $M$, fix a point $x\in M$ and let<br />
$$t_M(n) = vol B(x,n),$$<br />
where $B(x,r)$ is the ball on $M$ of radius $r$ and centre $x$ while $vol$<br />
is the measure on $M$ induced by the Riemannian structure. If $x'$ is another point of $M$, then $B(x',r)\subset B(x, r + r_0)$, where $r_0 =<br />
d(x, x')$ and $d$ is the distance function on $M$.<br />
Therefore, the following definition is correct.<br />
<br />
{{beginthm|Definition}} $gr (M) = [t_M]$ is called the '''growth type''' of a Riemannian manifold $M$. <br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} (\cite{Milnor1968a}, \cite{vSvarc1955}) If $M$ is compact, then the fundamental group $\pi_1 (M)$ and the universal covering $\tilde M$ of $M$ have the same growth type. If $M$ is a compact Riemannian manifold of negative sectional<br />
curvature, then both $\tilde M$ and $\pi_1(M)$ have exponential growth.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T14:55:21Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have $gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} Any finitely generated abelian group of rank $d$ has the growth type $[n^d]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem}} Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth \cite{Wolf1968}. Conversely, any finitely group of polynomial type of growth contains a nilpotent subgroup of finite index \cite{Gromov1981}.<br />
{{endthm}}<br />
<br />
Natural examples of finitely generated groups are provided by<br />
fundamental groups of compact manifolds. Below we show how the growth type<br />
of the fundamental group of a compact Riemannian manifold is related to<br />
its geometry. To this end let us consider a<br />
complete Riemannian manifold $M$, fix a point $x\in M$ and let<br />
$$t_M(n) = vol B(x,n),$$<br />
where $B(x,r)$ is the ball on $M$ of radius $r$ and centre $x$ while $vol$<br />
is the measure on $M$ induced by the Riemannian structure. If $x'$ is another point of $M$, then $B(x',r)\subset B(x, r + r_0)$, where $r_0 =<br />
d(x, x')$ and $d$ is the distance function on $M$.<br />
Therefore, the following definition is correct.<br />
<br />
{{beginthm|Definition}} $gr (M) = [t_M]$ is called the '''growth type''' of a Riemannian manifold $M$. <br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} (\cite{Milnor1968a}, \cite{vSvarc1955}) If $M$ is compact, then the fundamental group $\pi_1 (M)$ and the universal covering $\tilde M$ of $M$ have the same growth type.<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T14:45:54Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have $gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} Any finitely generated abelian group of rank $d$ has the growth type $[n^d]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem}} Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth \cite{Wolf1968}. Conversely, any finitely group of polynomial type of growth contains a nilpotent subgroup of finite index \cite{Gromov1981}.<br />
{{endthm}}<br />
<br />
Natural examples of finitely generated groups are provided by<br />
fundamental groups of compact manifolds. Below we show how the growth type<br />
of the fundamental group of a compact Riemannian manifold is related to<br />
its geometry. To this end let us consider a<br />
complete Riemannian manifold $M$, fix a point $x\in M$ and let<br />
$$t_M(n) = vol B(x,n),$$<br />
where $B(x,r)$ is the ball on $M$ of radius $r$ and centre $x$ while $vol$<br />
is the measure on $M$ induced by the Riemannian structure. If $x'$ is another point of $M$, then $B(x',r)\subset B(x, r + r_0)$, where $r_0 =<br />
d(x, x')$ and $d$ is the distance function on $M$.<br />
Therefore, the following definition is correct.<br />
<br />
{{beginthm|Definition}} $gr (M) = [t_M]$ is called the '''growth type''' of a Riemannian manifold $M$. <br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} If $M$ is compact, then the fundamental group $\pi_1 (M)$ and the universal covering $\tilde M$ of $M$ have the same growth type.<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T14:36:14Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have $gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} Any finitely generated abelian group of rank $d$ has the growth type $[n^d]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem}} Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth \cite{Wolf1968}. Conversely, any finitely group of polynomial type of growth contains a nilpotent subgroup of finite index \cite{Gromov1981}.<br />
{{endthm}}<br />
<br />
Natural examples of finitely generated groups are provided by<br />
fundamental groups of compact manifolds. Below we show how the growth type<br />
of the fundamental group of a compact Riemannian manifold is related to<br />
its geometry. To this end let us consider a<br />
complete Riemannian manifold $M$, fix a point $x\in M$ and let<br />
$$t_M(n) = vol B(x,n),$$<br />
where $B(x,r)$ is the ball on $M$ of radius $r$ and centre $x$ while $vol$<br />
is the measure on $M$ induced by the Riemannian structure. If $x'$ is another point of $M$, then $B(x',r)\subset B(x, r + r_0)$, where $r_0 =<br />
d(x, x')$ and $d$ is the distance function on $M$.<br />
Therefore, the following definition is correct.<br />
<br />
{{beginthm|Definition}} $gr (M) = [t_M]$ is called the '''growth type''' of a Riemannian manifold $M$. <br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T14:29:14Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have $gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} Any finitely generated abelian group of rank $d$ has the growth type $[n^d]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem}} Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth \cite{Wolf1968}. Conversely, any finitely group of polynomial type of growth contains a nilpotent subgroup of finite index \cite{Gromov1981}.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T14:28:52Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have $gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} Any finitely generated abelian group of rank $d$ has the growth type $[n^d]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem}} Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth \cite{Wolf1968}. Conversely, any finitely group of polynomial type of growth contains a nilpotent subgroup of finite index \cite(Gromov1981}.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T14:24:36Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
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{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have $gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} Any finitely generated abelian group of rank $d$ has the growth type $[n^d]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Theorem}} Any finitely generated nilpotent group is of polynomial type of growth. Moreover, if a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth \cite{Wolf1968}. Conversely, any finitely group of polynomial type of growth contains a nilpotent subgroup of finite index.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T14:21:00Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have $gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} Any finitely generated abelian group of rank $d$ has the growth type $[n^d]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} \cite{Wolf1968} Any finitely generated nilpotent group is of polynomial type of growth. Moreover, f a finitely generated group contains a nilpotent group of finite index, then it has a polynomial type of growth.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T14:19:17Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have $gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} Any finitely generated abelian group of rank $d$ has the growth type $[n^d]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} \cite{Wolf1968} Any finitely generated nilpotent group is of polynomial type of growth.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T12:16:18Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have $gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} Any finitely generated abelian group of rank $d$ has the growth type $[n^d]$.<br />
\{{endthm}}<br />
<br />
{{beginthm|Proposition}} Any finitely generated nilpotent group is of polynomial type of growth.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T12:12:20Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have<br />
$gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T12:10:01Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have<br />
$gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
<br />
More generally, let<br />
$$<br />
\begin{CD}<br />
1 @>>> H' @>>> H @>>> H/H' @>>> 1 \\<br />
@. @VVV @VV\iota V @VVV @.\\<br />
1 @>>> G' @>>> G @>>> G/G' @>>> 1<br />
\end{CD}<br />
$$<br />
be a commutative diagram of finitely generated groups with $G' = \iota (H')$,<br />
$\iota :H\to G$ being a homomorphism.<br />
<br />
{{beginthm|Proposition}} If $\iota$ is a monomorphism, then<br />
$\gr (H/H')\preceq\gr (G/G')$. If, moreover, $\iota (H)$ has finite index<br />
in $G$, then $\gr (H/H') = \gr (G/G')$.<br />
{{endthm}}<br />
<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T12:06:54Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}} For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have<br />
$gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T12:06:15Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}}For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have<br />
$gr (G/H)\preceq gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T12:05:33Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
<br />
{{beginthm|Proposition}}For any finitely generated group $G$ and<br />
any normal subgroup $H$ of $G$ we have<br />
$\gr (G/H)\preceq\gr (G)\preceq [e^n].$<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T12:02:36Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}}A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T12:02:18Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
For most of results listed here we refer to \cite{Hector&Hirsch1981}.<br />
<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}}A finite group has the growth type $[1]$ while the<br />
abelian group $\Bbb Z^d$ has the polynomial growth of degree $d$. Any free<br />
(non-abelian) group has the exponential growth $[e^n]$.<br />
{{endthem}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T11:56:21Z<p>Pawel Walczak: /* Growth in groups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
Let $G$ be a finitely generated group and $G_1$ a finite {\it symmetric}<br />
(i.e. such that $e\in G_1$ and $G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1$)<br />
set generating it. For any $n\in\Bbb N$ let<br />
$$<br />
G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}<br />
$$<br />
and<br />
$$t_G(n) = \# G_n.$$<br />
The type of growth of $t_G$ does not depend on $G_1$, so we may write the following.<br />
<br />
{{beginthm|Definition}} The '''growth type'''<br />
$gr (G)$ of $G$ is defined as $[t_G]$ for any finite symmetric generating set $G_1$. If $G$<br />
acts on a space $X$ and $x\in X$, then the '''growth type''' $gr (G,x)$ of $G$ at $x$<br />
is defined in a similar way: $gr (G,x) = [t_x]$, where<br />
$$t_x(n) = \#\{ g(x);\ g\in G_n\}$$<br />
for any fixed finite symmetric generating set $G_1$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T11:31:35Z<p>Pawel Walczak: /* Growth */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
'''polynomial''' (of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called '''exponential'''.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T11:30:25Z<p>Pawel Walczak: /* Growth */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
<br />
{{beginthm|Example}} $[0]\preceq [1]\preceq [\log n]\preceq [n]\preceq<br />
[n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n,<br />
3^n,\dots )]$ and all the growth types listed above are different. The growth<br />
type of any polynomial of degree $d\ge 0$ is equal to $[n^d]$ and is called<br />
{\it polynomial}(of degree $d$). $[a^n] = [e^n]$<br />
for any $a > 1$ and this growth type is called {\it exponential}.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T11:28:17Z<p>Pawel Walczak: /* Expansion growth */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T11:27:35Z<p>Pawel Walczak: /* Growth */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
</wikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
<\endwikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T11:26:16Z<p>Pawel Walczak: /* Growth */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
===Growth in groups===<br />
<wikitex>;<br />
<br />
<\endwikitex><br />
<br />
<br />
===Orbit growth in pseudogroups===<br />
<wikitex>;<br />
<br />
<\endwikitex><br />
<br />
<br />
===Expansion growth===<br />
<wikitex>;<br />
<br />
<\endwikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T11:20:59Z<p>Pawel Walczak: /* Growth */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{Hector&Hirsch1981}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T11:19:41Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{HH}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{{endthm}}<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T11:18:21Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{{beginthm|Definition}} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{HH}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
{(endthm}}<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T11:17:19Z<p>Pawel Walczak: /* Growth */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
{beginthm|Definition} Elements of the quotient $\hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq$ are called<br />
'''growth types'''. The growth type of $(t_j)\in\hat{\mathcal{T}}$<br />
(resp., of $t\in\mathcal{T}$) is denoted by $[(t_j)]$ (resp., by $[t]$). Also,<br />
we let $\mathcal{E} = \{ [t];\, t\in\mathcal{T}\}$. $\mathcal{E}$ is the set of growth<br />
types of monotone functions (in the sense of \cite{HH}). The preorder<br />
$\preceq$ induces a partial order (denoted again by $\preceq$) in<br />
$\hat{\mathcal{E}}$.<br />
(endthm}<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T11:10:35Z<p>Pawel Walczak: /* Growth */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
(*) $(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
induces an equivalence relation $\simeq$ in $\hat{\mathcal{T}}$:<br />
$$<br />
(t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\<br />
\text{and}\ (\tau_k)\preceq (t_j).<br />
$$<br />
In particular, if $t$ and $\tau\in\mathcal{T}$, then<br />
$$<br />
t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)<br />
$$<br />
and<br />
$$<br />
t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le<br />
t(n)\le a\tau (bn)<br />
$$<br />
for all $n\in\Bbb N$ and some $a,b\in\Bbb N$. (Here, $[x]$ is the largest<br />
integer which does not exceed $x$, $x\in\Bbb R$.)<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T11:05:10Z<p>Pawel Walczak: /* Growth */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
Let now $\mathcal{T}$ be the set of non-negative increasing functions defined on<br />
$\Bbb N$:<br />
$$<br />
\mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\<br />
n\in\Bbb N\} ,<br />
$$<br />
and $\hat{\mathhcal{T}}$ the set of increasing sequences with entries in $\mathcal{T}$:<br />
$$<br />
\hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le<br />
t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .<br />
$$<br />
Elements $t$ of $\mathcal{T}$ can be identified with sequences $(t,t,\dots )$, so<br />
$\mathcal{T}$ can be considered as a subset of $\hat{\mathcal{T}}$.<br />
<br />
A preorder $\preceq$ defined by the condition:<br />
<br />
$(t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N}$ if and only<br />
if there exists $b\in\Bbb N$ such that for all $j\in\Bbb N$ the inequalities<br />
$t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,$<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T10:57:25Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $c$ such that the inequalities<br />
$$a_n\le a\cdot b_{cn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T10:55:44Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
Let us begin with two non-decreasing sequences $(a_n)$ and $(b_n)$ of<br />
non-negative numbers. We shall say that $(a_n)$ "grows slower" that<br />
$(b_n)$ ($(a_n)\preceq (b_n)$) whenever there exist positive constants $a$<br />
and $b$ such that the inequalities<br />
$$a_n\le a\cdot b_{Bn}$$<br />
hold for all $n\in\mathbb N$. We say that '''types of growth'''<br />
of our sequences $(a_n)$ and $(b_n)$ are the same whenever<br />
$$(a_n)\preceq (b_n)\preceq (a_n).$$<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T10:00:40Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whitehead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T09:59:03Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whithead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T09:56:46Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in \cite{Veblen&Whithead1932} in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Andr\'e Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T09:55:17Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the<br />
first time in the paper by Veblen and Whitehead \cite{Veblen&Whithead1932} in the context of geometric objects studied in differential geometry.Pseudogroups where brought into the foliation theory by Andr\'e Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T09:48:48Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. Pseudogroups where invented by Ehresmann anmd brought into the foliation theory by Andr\'e Haefliger \cite{Haefliger1962a}. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T09:37:03Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. Pseudogroups where invented by Ehresmann anmd brought into the foliation theory by Andr\'e Haefliger. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys&Langevin&Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-27T09:34:47Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy (see [[Foliations#Holonomy]]) pseudogroup. Pseudogroups where invented by Ehresmann anmd brought into the foliation theory by Andr\'e Haefliger. The growth types of leaves (equivalent to the growth types of the corresponding orbits of holonomy pseudogroups) as well as the expansion growth of a foliation (or, its holonomy pseudogroup) describe some aspects of the foliation dynamics. Corresponding to the expansion growth is the notion of geometric entropy of a foliation \cite{Ghys@Langevin@Walczak1988}. Vanishing entropy can be seen as a dynamical condition which occurs to have strong geometric and topological consequences (see [[Dynamics of foliations#Results on entropy]]) below. <br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-26T18:14:17Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy psudogroup (see [[Foliations#Holonomy]]).<br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any element of $\mathcal{H}$ assigns to a plaque $P$ the end plaque $P'$ of a '''chain''' (that is a sequence of plaques such that any two consequtive plaques intersect) which oroginates at $P$. <br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-26T18:11:00Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy psudogroup (see [[Foliations#Holonomy]]).<br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
This means that any holonomy map assigns to a plaque $P$ the end plaque $P'$ of a '''chain''', that is a sequence of plaques origanet at $P$ and such that any two consequtive plaques intersect. <br />
<br />
<br />
Certainly, holonomynomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/File:Chain1.jpegFile:Chain1.jpeg2010-11-26T18:06:50Z<p>Pawel Walczak: </p>
<hr />
<div></div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-26T18:06:04Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy psudogroup (see [[Foliations#Holonomy]]).<br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain1.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
<br />
Certainly, holonomynomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-26T18:04:12Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy psudogroup (see [[Foliations#Holonomy]]).<br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
<br />
Certainly, holonomynomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-26T18:03:36Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy psudogroup (see [[Foliations#Holonomy]]).<br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
Certainly, holonomynomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/File:Chain.jpegFile:Chain.jpeg2010-11-26T17:55:51Z<p>Pawel Walczak: </p>
<hr />
<div></div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-26T17:54:41Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy psudogroup (see [[Foliations#Holonomy]]).<br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
[[Image:chain.jpeg|thumb|300px|Chain of plaques]]<br />
<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-26T17:16:37Z<p>Pawel Walczak: /* Holonomy pesudogroups */</p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy psudogroup (see [[Foliations#Holonomy]]).<br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-26T17:14:19Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy psudogroup (see [[Foliations#Holonomy]]).<br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times\Bbb R$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczakhttp://www.map.mpim-bonn.mpg.de/Dynamics_of_foliationsDynamics of foliations2010-11-26T17:12:18Z<p>Pawel Walczak: </p>
<hr />
<div><!-- COMMENT: <br />
<br />
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:<br />
<br />
- For statements like Theorem, Lemma, Definition etc., use e.g.<br />
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.<br />
<br />
- For references, use e.g. {{cite|Milnor1958b}}.<br />
<br />
END OF COMMENT<br />
<br />
-->{{Authors|Pawel Walczak}}<br />
{{Stub}}<br />
== Introduction ==<br />
<wikitex>;<br />
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits \cite{Walczak2004}. The dynamics of a foliation can be described in terms of its holonomy psudogroup (see [[Foliations#Holonomy]]).<br />
</wikitex><br />
<br />
<br />
<br />
== Pseudogroups ==<br />
<wikitex>;<br />
The notion of a pseudogroup generalizes that of a group of<br />
transformations. Given a space $X$, any group of transformations of $X$<br />
consists of maps defined globally on $X$, mapping $X$ bijectively onto<br />
itself and such that the composition of any two of them as well as the<br />
inverse of any of them belongs to the group. The same holds for a<br />
pseudogroup with this difference that the maps are not defined globally<br />
but on open subsets, so the domain of the composition is usually smaller<br />
than those of the maps being composed.<br />
<br />
To make the above precise, let us take a topological space $X$ and denote<br />
by Homeo$\, (X)$ the family of all<br />
homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then<br />
$D_g$ is its domain and $R_g = g(D_g)$.<br />
<br />
{{beginthm|Definition}}<br />
A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup'''<br />
if it<br />
is closed under composition, inversion, restriction to open subdomains and<br />
unions. More precisely, $\Gamma$ should satisfy the following conditions:<br />
<br />
(i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$,<br />
<br />
(ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$,<br />
<br />
(iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is<br />
open,<br />
<br />
(iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and<br />
$g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$.<br />
<br />
Moreover, we shall always assume that<br />
<br />
(v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$).<br />
<br />
\label{def:psgroup}{{endthm}}<br />
<br />
<br />
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, $C^k$, $k = 1, 2, \ldots, \infty, \omega$, between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group $G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is '''generated''' by $G$.)<br />
Any set $A$ of homeomorphisms bewteen open sets (with domains covering a space $X$) generates ma pseudogroup $\Gamma (A)$ which is the smallest pseudogroup containing $A$; precisely a homeomorphism $h:D_h\to R_h$ belongs to $\Gamma (A)$ if and only if for any point $x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$ of $x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$. If $A$ is finite, $\Gamma (A)$ is said to be '''finitely generated'''.<br />
<br />
</wikitex><br />
<br />
==Holonomy pesudogroups==<br />
<wikitex>;<br />
<br />
{{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be<br />
'''nice''' (also, '''nice''' is the<br />
covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if<br />
<br />
(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,<br />
<br />
(ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$<br />
is an open cube,<br />
<br />
(iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$,<br />
then there exists a foliated chart chart $\chi$ and such that<br />
$R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup<br />
D_\psi$ and $\phi = \chi |D_\phi$.<br />
{{endthm}}<br />
<br />
Since manifolds are supposed to be paracompact here, they are separable and<br />
hence nice coverings are denumerable. Nice coverings on compact manifolds are<br />
finite. On arbitrary foliated manfiolds, nice coverings do exist.<br />
<br />
Given a nice covering $\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$. For<br />
any $U\in\mathcal{U}$, let $T_U$ be the space of the '''plaques''' (i.e., connected components of intersections $U\cap L$. $L$ being a leaf of $\mathcal{F}$) of $\mathcal{F}$ contained in $U$. Equip $T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of<br />
$U$ are equivalent iff they belong to the same plaque. $T_U$ is homeomorphic<br />
($C^r$-diffeomorphic when $\mathcal{F}$ is $C^r$-differentiable and $r\ge 1$) to an<br />
open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map<br />
$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$.<br />
The disjoint union<br />
$$T = \sqcup\{T_U; U\in\mathcal{U}\}$$<br />
is called a '''complete transversal''' for $\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$ is differentiable of class $C^r$, $r > 0$, each of the spaces $T_U$ can be mapped homeomorphically onto a $C^r$-submanifold $T_U'\subset U$<br />
transverse to $U$: if $x\in T_U'$ and $L$ is the leaf of $\mathcal{F}$ passing through $x$,<br />
then<br />
$$T_xM = T_xT_U'\oplus T_xL.$$<br />
Completeness of $T$ means that every leaf of $\mathcal{F}$ intersects at least one<br />
of the submanifolds $T_U'$.<br />
<br />
{{beginthm|Definition}} Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$<br />
the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,<br />
$D_{VU}$ being the open subset of $T_U$ which consists of all the plaques<br />
$P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following<br />
way:<br />
$$<br />
h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\<br />
P\subset U\ \text{and}\ P'\subset V\<br />
\text{intersect}.<br />
$$<br />
All the maps $h_{UV}$ ($U, V\in\mathcal{U}$) generate a pseudogroup $\mathcal{H}$ on $T$.<br />
$\mathcal{H}$ is called the '''holonomy pseudogroup''' of $\mathcal{F}$.<br />
\label{def:holonomy}{{endthm}}<br />
<br />
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see [[Foliations#Reeb Foliations]]) on $D^2\times\Bbb R$ one gets a foliation of the 3-dimensional sphere $S^3$ for which any arc $T$ intersecting the unique toral leaf $T^2$ is a complete transversal; $T$ can be identified with a segment $(-\epsilon, \epsilon)$ ($\epsilon > 0$), the point of intersection $T\cap T^2$ with the number $0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$$ $h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$ contract their domains towards $0$. <br />
<br />
</wikitex><br />
<br />
==Growth==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Geometric entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Invariant measures==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
==Results on entropy==<br />
<wikitex>;<br />
<br />
<br />
</wikitex><br />
<br />
== References ==<br />
{{#RefList:}}<br />
<br />
[[Category:Theory]]</div>Pawel Walczak