# Dynamics of foliations

## 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 $X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}X$, any group of transformations of $X$$X$ consists of maps defined globally on $X$$X$, mapping $X$$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$$X$ and denote by Homeo$\, (X)$$\, (X)$ the family of all homeomorphisms between open subsets of $X$$X$. If $g\in$$g\in$ Homeo$\, (X)$$\, (X)$, then $D_g$$D_g$ is its domain and $R_g = g(D_g)$$R_g = g(D_g)$.

Definition 2.1. A subfamily $\Gamma$$\Gamma$ of Homeo$\, (X)$$\, (X)$ is said to be a pseudogroup if it is closed under composition, inversion, restriction to open subdomains and unions. More precisely, $\Gamma$$\Gamma$ should satisfy the following conditions:

(i) $g\circ h\in\Gamma$$g\circ h\in\Gamma$ whenever $g$$g$ and $h\in\Gamma$$h\in\Gamma$,

(ii) $g^{-1}\in\Gamma$$g^{-1}\in\Gamma$ whenever $g\in\Gamma$$g\in\Gamma$,

(iii) $g|U\in\Gamma$$g|U\in\Gamma$ whenever $g\in\Gamma$$g\in\Gamma$ and $U\subset D_g$$U\subset D_g$ is open,

(iv) if $g\in$$g\in$ Homeo$\, (X)$$\, (X)$, $\mathcal{U}$$\mathcal{U}$ is an open cover of $D_g$$D_g$ and $g|U\in\Gamma$$g|U\in\Gamma$ for any $U\in\mathcal{U}$$U\in\mathcal{U}$, then $g\in\Gamma$$g\in\Gamma$.

Moreover, we shall always assume that

(v] id$_X\in\Gamma$$_X\in\Gamma$ (or, equivalently, $\cup\{ D_g; g\in\Gamma\} = X$$\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$$C^k$, $k = 1, 2, \ldots, \infty, \omega$$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$$G$ of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is generated by $G$$G$.) Any set $A$$A$ of homeomorphisms bewteen open sets (with domains covering a space $X$$X$) generates ma pseudogroup $\Gamma (A)$$\Gamma (A)$ which is the smallest pseudogroup containing $A$$A$; precisely a homeomorphism $h:D_h\to R_h$$h:D_h\to R_h$ belongs to $\Gamma (A)$$\Gamma (A)$ if and only if for any point $x\in D_h$$x\in D_h$ there exist elements $h_1, \ldots, h_k\in A$$h_1, \ldots, h_k\in A$, exponents $\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$$\epsilon _1, \ldots, \epsilon _k\in\{ \pm 1\}$ and a neighbourhood $V$$V$ of $x$$x$ such that $h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$$h = h_1^{\epsilon _1}\circ\ldots\circ h_k^{\epsilon _k}$ on $V$$V$. If $A$$A$ is finite, $\Gamma (A)$$\Gamma (A)$ is said to be finitely generated.

## 3 Holonomy pesudogroups

Definition 3.1. A foliated atlas $\mathcal{A}$$\mathcal{A}$ on a foliated manifold $(M, \mathcal{F})$$(M, \mathcal{F})$ is said to be nice (also, nice is the covering of $M$$M$ by the domains $D_\phi$$D_\phi$ of the charts $\phi\in\mathcal{A}$$\phi\in\mathcal{A}$) if

(i) the covering $\{ D_\phi ; \phi\in\mathcal{A}\}$$\{ D_\phi ; \phi\in\mathcal{A}\}$ is locally finite,

(ii) for any $\phi\in\mathcal{A}$$\phi\in\mathcal{A}$, $R_\phi = \phi (D_\phi )\subset\Bbb R^n$$R_\phi = \phi (D_\phi )\subset\Bbb R^n$ is an open cube,

(iii) if $\phi$$\phi$ and $\psi\in\mathcal{A}$$\psi\in\mathcal{A}$, and $D_\phi\cap D_\psi\ne\emptyset$$D_\phi\cap D_\psi\ne\emptyset$, then there exists a foliated chart chart $\chi$$\chi$ and such that $R_\chi$$R_\chi$ is an open cube, $D_\chi$$D_\chi$ contains the closure of $D_\phi\cup D_\psi$$D_\phi\cup D_\psi$ and $\phi = \chi |D_\phi$$\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}$$\mathcal{U}$ of a foliated manifold $(M, \mathcal{F})$$(M, \mathcal{F})$. For any $U\in\mathcal{U}$$U\in\mathcal{U}$, let $T_U$$T_U$ be the space of the plaques (i.e., connected components of intersections $U\cap L$$U\cap L$. $L$$L$ being a leaf of $\mathcal{F}$$\mathcal{F}$) of $\mathcal{F}$$\mathcal{F}$ contained in $U$$U$. Equip $T_U = U/(\mathcal{F}|U)$$T_U = U/(\mathcal{F}|U)$ with the quotient topology: two points of $U$$U$ are equivalent iff they belong to the same plaque. $T_U$$T_U$ is homeomorphic ($C^r$$C^r$-diffeomorphic when $\mathcal{F}$$\mathcal{F}$ is $C^r$$C^r$-differentiable and $r\ge 1$$r\ge 1$) to an open cube $Q\subset\Bbb R^q$$Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$$q = \text{codim}\, \mathcal{F}$) via the map $\phi''$$\phi''$, where $\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$$\phi = (\phi ', \phi ''):U\to\Bbb R^p\times\Bbb R^q$ is a foliated chart on $U$$U$. The disjoint union

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

is called a complete transversal for $\mathcal{F}$$\mathcal{F}$. Transversality refers to the fact that, if $\mathcal{F}$$\mathcal{F}$ is differentiable of class $C^r$$C^r$, $r > 0$$r > 0$, each of the spaces $T_U$$T_U$ can be mapped homeomorphically onto a $C^r$$C^r$-submanifold $T_U'\subset U$$T_U'\subset U$ transverse to $U$$U$: if $x\in T_U'$$x\in T_U'$ and $L$$L$ is the leaf of $\mathcal{F}$$\mathcal{F}$ passing through $x$$x$, then

$\displaystyle T_xM = T_xT_U'\oplus T_xL.$

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

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

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$$D^2\times S^1$ one gets a foliation of the 3-dimensional sphere $S^3$$S^3$ for which any arc $T$$T$ intersecting the unique toral leaf $T^2$$T^2$ is a complete transversal; $T$$T$ can be identified with a segment $(-\epsilon, \epsilon)$$(-\epsilon, \epsilon)$ ($\epsilon > 0$$\epsilon > 0$), the point of intersection $T\cap T^2$$T\cap T^2$ with the number $0$$0$, while the holonomy group with the one on our segment generated by two maps $h_+, h_- :T\to T$$h_+, h_- :T\to T$ such that $h_+|[0, \epsilon ) = \id$$h_+|[0, \epsilon ) = \id$, $h_-|(-\epsilon , 0] = \id$$h_-|(-\epsilon , 0] = \id$, $h_+|(-\epsilon , 0]$$h_+|(-\epsilon , 0]$ snd $h_-|[0, \epsilon)$$h_-|[0, \epsilon)$ contract their domains towards $0$$0$.