Dynamics of foliations

Revision as of 18:16, 26 November 2010

1 Introduction

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 [Walczak2004]. The dynamics of a foliation can be described in terms of its holonomy psudogroup (see Foliations#Holonomy).

2 Pseudogroups

The notion of a pseudogroup generalizes that of a group of transformations. Given a space $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT -->{{Authors|Pawel Walczak}} {{Stub}} == Introduction == <wikitex>; 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]]). </wikitex> == Pseudogroups == <wikitex>; The notion of a pseudogroup generalizes that of a group of transformations. Given a space $X$, any group of transformations of $X$ consists of maps defined globally on $X$, mapping $X$ bijectively onto itself and such that the composition of any two of them as well as the inverse of any of them belongs to the group. The same holds for a pseudogroup with this difference that the maps are not defined globally but on open subsets, so the domain of the composition is usually smaller than those of the maps being composed. To make the above precise, let us take a topological space $X$ and denote by Homeo$\, (X)$ the family of all homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then $D_g$ is its domain and $R_g = g(D_g)$. {{beginthm|Definition}} A subfamily $\Gamma$ of Homeo$\, (X)$ is said to be a '''pseudogroup''' if it is closed under composition, inversion, restriction to open subdomains and unions. More precisely, $\Gamma$ should satisfy the following conditions: (i) $g\circ h\in\Gamma$ whenever $g$ and $h\in\Gamma$, (ii) $g^{-1}\in\Gamma$ whenever $g\in\Gamma$, (iii) $g|U\in\Gamma$ whenever $g\in\Gamma$ and $U\subset D_g$ is open, (iv) if $g\in$ Homeo$\, (X)$, $\mathcal{U}$ is an open cover of $D_g$ and $g|U\in\Gamma$ for any $U\in\mathcal{U}$, then $g\in\Gamma$. Moreover, we shall always assume that (v] id$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$). \label{def:psgroup}{{endthm}} 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 $G$.) 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'''. </wikitex> ==Holonomy pesudogroups== <wikitex>; {{beginthm|Definition}} A foliated atlas $\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$ is said to be '''nice''' (also, '''nice''' is the covering of $M$ by the domains $D_\phi$ of the charts $\phi\in\mathcal{A}$) if (i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite, (ii) for any $\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$ is an open cube, (iii) if $\phi$ and $\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$, then there exists a foliated chart chart $\chi$ and such that $R_\chi$ is an open cube, $D_\chi$ contains the closure of $D_\phi\cup D_\psi$ and $\phi = \chi |D_\phi$. {{endthm}} Since manifolds are supposed to be paracompact here, they are separable and hence nice coverings are denumerable. Nice coverings on compact manifolds are finite. On arbitrary foliated manfiolds, nice coverings do exist. {{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 $U\cap V\ne\emptyset$ the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$, $D_{VU}$ being the open subset of $T_U$ which consists of all the plaques $P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following way: \begin{equation} h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\ P\subset U\ \text{and}\ P'\subset V\ \text{intersect}. \label{eq:holonomy}\end{equation} \label{def:holonomy}{{endthm}} </wikitex> ==Growth== <wikitex>; </wikitex> ==Geometric entropy== <wikitex>; </wikitex> ==Invariant measures== <wikitex>; </wikitex> ==Results on entropy== <wikitex>; </wikitex> == References == {{#RefList:}} [[Category:Theory]]X$, any group of transformations of $X$ consists of maps defined globally on $X$, mapping $X$ bijectively onto itself and such that the composition of any two of them as well as the inverse of any of them belongs to the group. The same holds for a pseudogroup with this difference that the maps are not defined globally but on open subsets, so the domain of the composition is usually smaller than those of the maps being composed.

To make the above precise, let us take a topological space $X$ and denote by Homeo$\, (X)$ the family of all homeomorphisms between open subsets of $X$. If $g\in$ Homeo$\, (X)$, then $D_g$ is its domain and $R_g = g(D_g)$.

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 $G$.) 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.

3 Holonomy pesudogroups

Since manifolds are supposed to be paracompact here, they are separable and hence nice coverings are denumerable. Nice coverings on compact manifolds are finite. On arbitrary foliated manfiolds, nice coverings do exist.

Definition 3.2. Given a nice covering $\mathcal{U}$ of a foliated manifold $(M,\mathcal{F})$ and two sets $U$ and $V\in\mathcal{U}$ such that $U\cap V\ne\emptyset$ the holonomy map $h_{VU}: D_{VU}\to T_V$, $D_{VU}$ being the open subset of $T_U$ which consists of all the plaques $P$ of $U$ for which $P\cap V\ne\emptyset$, is defined in the following way:

(1)$h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\ P\subset U\ \text{and}\ P'\subset V\ \text{intersect}. <label>eq:holonomy</label>$

4 Growth

5 Geometric entropy

6 Invariant measures

7 Results on entropy

8 References

• [Walczak2004] P. Walczak, Dynamics of foliations, groups and pseudogroups, Birkhäuser Verlag, 2004. MR2056374 (2005d:57042) Zbl 1084.37022