Dynamics of foliations

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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 (see Foliations#Holonomy) pseudogroup. As far as we know, the notion of a pseudogroup appeared for the first time in [Veblen&Whitehead1932] in the context of geometric objects studied in differential geometry. Pseudogroups have been brought into the foliation theory by Haefliger [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 [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.

2 Pseudogroups

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).

Definition 2.1. 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).

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

Definition 3.1. 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.

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.

Given a nice covering \mathcal{U} of a foliated manifold (M, \mathcal{F}). For 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 U are equivalent iff they belong to the same plaque. T_U is homeomorphic (C^r-diffeomorphic when \mathcal{F} is C^r-differentiable and r\ge 1) to an open cube Q\subset\Bbb R^q (q = \text{codim}\, \mathcal{F}) via the map \phi'', where \phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q is a foliated chart on U. The disjoint union

\displaystyle T = \sqcup\{T_U; U\in\mathcal{U}\}

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 transverse to U: if x\in T_U' and L is the leaf of \mathcal{F} passing through x, then

\displaystyle T_xM = T_xT_U'\oplus T_xL.

Completeness of T means that every leaf of \mathcal{F} intersects at least one of the submanifolds T_U'.

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:

\displaystyle  h_{VU}(P) = P'\ \Leftrightarrow\ \text{the plaques}\ P\subset U\ \text{and}\ P'\subset V\ \text{intersect}.

All the maps h_{UV} (U, V\in\mathcal{U}) generate a pseudogroup \mathcal{H} on T. \mathcal{H} is called the holonomy pseudogroup of \mathcal{F}.

Chain of plaques

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.

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.

4 Growth

Let us begin with two non-decreasing sequences (a_n) and (b_n) of non-negative numbers. We shall say that (a_n) "grows slower" that (b_n) ((a_n)\preceq (b_n)) whenever there exist positive constants a and c such that the inequalities

\displaystyle a_n\le a\cdot b_{cn}

hold for all n\in\mathbb N. We say that types of growth of our sequences (a_n) and (b_n) are the same whenever

\displaystyle (a_n)\preceq (b_n)\preceq (a_n).

Let now \mathcal{T} be the set of non-negative increasing functions defined on \Bbb N:

\displaystyle  \mathcal{T} = \{ t:\Bbb N\to [0,\infty ); t(n)\le t(n+1)\ \text{for all}\ n\in\Bbb N\} ,

and \hat{\mathcal{T}} the set of increasing sequences with entries in \mathcal{T}:

\displaystyle  \hat{\mathcal{T}} = \{ (t_j)_{j\in\Bbb N}; t_j\in\mathcal{T}\ \text{and}\ t_j(n)\le t_{j+1}(n)\ \text{for all}\ j\ \text{and}\ n\in\Bbb N\} .

Elements t of \mathcal{T} can be identified with sequences (t,t,\dots ), so \mathcal{T} can be considered as a subset of \hat{\mathcal{T}}.

A preorder \preceq defined by the condition:

(*) (t_j)_{j\in\Bbb N}\preceq (\tau_k)_{k\in\Bbb N} if and only if there exists b\in\Bbb N such that for all j\in\Bbb N the inequalities t_j(n)\le a\tau_k (bn), \quad n\in\Bbb N,

induces an equivalence relation \simeq in \hat{\mathcal{T}}:

\displaystyle  (t_j)\simeq (\tau_k)\ \text{if and only if}\ (t_j)\preceq (\tau_k)\ \text{and}\ (\tau_k)\preceq (t_j).

In particular, if t and \tau\in\mathcal{T}, then

\displaystyle  t\preceq\tau\Leftrightarrow t(n)\le a\tau (bn)


\displaystyle  t\simeq\tau\Leftrightarrow a^{-1}\tau ([n/b] )\le t(n)\le a\tau (bn)

for all n\in\Bbb N and some a,b\in\Bbb N. (Here, [x] is the largest integer which does not exceed x, x\in\Bbb R.)

Definition 4.1. Elements of the quotient \hat{\mathcal{E}} = \hat{\mathcal{T}}/\simeq are called growth types. The growth type of (t_j)\in\hat{\mathcal{T}} (resp., of t\in\mathcal{T}) is denoted by [(t_j)] (resp., by [t]). Also, we let \mathcal{E} = \{ [t];\, t\in\mathcal{T}\}. \mathcal{E} is the set of growth types of monotone functions (in the sense of [Hector&Hirsch1981]). The preorder \preceq induces a partial order (denoted again by \preceq) in \hat{\mathcal{E}}.

Example 4.2. [0]\preceq [1]\preceq [\log n]\preceq [n]\preceq [n^2]\preceq\dots\preceq [(1, n, n^2, \dots )]\preceq [2^n]\preceq [(1, 2^n, 3^n,\dots )] and all the growth types listed above are different. The growth type of any polynomial of degree d\ge 0 is equal to [n^d] and is called polynomial (of degree d). [a^n] = [e^n] for any a > 1 and this growth type is called exponential.

4.1 Growth in groups

For most of results listed here we refer to [Hector&Hirsch1981].

Let G be a finitely generated group and G_1 a finite symmetric (i.e. such that e\in G_1 and G_1^{-1} = \{ g^{-1};\, g\in G_1\}\subset G_1) set generating it. For any n\in\Bbb N let

\displaystyle  G_n = \{ g_1\cdot\dots\cdot g_n;\ g_1,\dots , g_n\in G_1\}


\displaystyle t_G(n) = \# G_n.

The type of growth of t_G does not depend on G_1, so we may write the following.

Definition 4.3. The growth type gr (G) of G is defined as [t_G] for any finite symmetric generating set G_1. If G acts on a space X and x\in X, then the growth type gr (G,x) of G at x is defined in a similar way: gr (G,x) = [t_x], where

\displaystyle t_x(n) = \#\{ g(x);\ g\in G_n\}

for any fixed finite symmetric generating set G_1.

Example 4.4. A finite group has the growth type [1] while the abelian group \Bbb Z^d has the polynomial growth of degree d. Any free (non-abelian) group has the exponential growth [e^n].

Proposition 4.5. For any finitely generated group G and any normal subgroup H of G we have gr (G/H)\preceq gr (G)\preceq [e^n].

Proposition 4.6. Any finitely generated abelian group of rank d has the growth type [n^d].

Theorem 4.7. 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 [Wolf1968]. Conversely, any finitely generated group of polynomial type of growth contains a nilpotent subgroup of finite index [Gromov1981].

Natural examples of finitely generated groups are provided by fundamental groups of compact manifolds. Below we show how the growth type of the fundamental group of a compact Riemannian manifold is related to its geometry. To this end let us consider a complete Riemannian manifold M, fix a point x\in M and let

\displaystyle t_M(n) = vol B(x,n),

where B(x,r) is the ball on M of radius r and centre x while vol 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 = d(x, x') and d is the distance function on M. Therefore, the following definition is correct.

Definition 4.8. gr (M) = [t_M] is called the growth type of a Riemannian manifold M.

Proposition 4.9 ([Milnor1968a], [Švarc1955]). 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 curvature, then both \tilde M and \pi_1(M) have exponential growth.

Fundamental groups of compact manifolds of negative sectional curvature are hyperbolic in the sense of [Gromov1987]. The last statement of the Proposition above can be generalized as follows,

Proposition 4.10. Any non-elementary hyperbolic group has exponential growth.

4.2 Orbit growth in pseudogroups

4.3 Expansion growth

5 Geometric entropy

6 Invariant measures

7 Results on entropy

8 References

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