Dynamics of foliations

From Manifold Atlas
Revision as of 16:17, 26 November 2010 by Pawel Walczak (Talk | contribs)
Jump to: navigation, search

The user responsible for this page is Pawel Walczak. No other user may edit this page at present.

This page has not been refereed. The information given here might be incomplete or provisional.

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, 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 Template:Generated by $G$.)

3 References

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox