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 psudogroup (see Foliations#Holonomy).

2 Pseudogroups

The notion of a pseudogroup generalizes that of a group of transformations. Given a space $X$$\newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}X$, any group of transformations of $X$$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 {\bold 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$).

3 References

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