Dynamics of Foliations, Groups and Pseudogroups

This book deals with the dynamics of general systems such as foliations, groups and pseudogroups, systems which are closely related via the notion of holonomy. It concentrates on notions and results related to different ways of measuring complexity of systems under consideration. More precisely, it deals with different types of growth, entropies and dimensions of limiting objects. Problems related to the topics covered are provided throughout the book.

Contenu

1 Dynamical systems.- 1.1 Pseudogroups.- 1.2 First examples.- 1.3 Foliations, laminations and holonomy.- 1.4 Markov pseudogroups.- 1.5 Hyperbolic spaces and groups.- 2 Growth.- 2.1 Growth types.- 2.2 Growth in groups.- 2.3 Orbit growth for pseudogroups.- 2.4 Expansion growth.- 3 Entropy.- 3.1 Entropy of classical systems.- 3.1.1 Topological entropy of a transformation.- 3.1.2 Invariant measures.- 3.1.3 Measure-theoretic entropy.- 3.1.4 Examples.- 3.1.5 Variational principle.- 3.2 Entropy of pseudogroups.- 3.3 Geometric entropy of foliations.- 3.4 Relating various entropies.- 3.5 Examples and constructions.- 3.5.1 Pullback.- 3.5.2 Gluing.- 3.5.3 Turbulizat ion.- 3.6 Entropy and resiliency.- 4 Invariant measures.- 4.1 Basic definitions and facts.- 4.2 Transverse invariant measures and homology.- 4.3 Measures and orbit growth.- 4.4 Transverse invariant measures in codimension 1.- 4.5 Vanishing entropy and invariant measures.- 4.6 Entropy, geodesic flow and invariant measures.- 4.7 Harmonic measures.- 4.8 Patterson-Sullivan measures.- 5 Hausdorff dimension.- 5.1 Definitions and basic facts.- 5.2 Julia sets.- 5.3 Dimension in foliated manifolds.- 5.4 Dimension of a hyperbolic boundary.- 5.5 Dimension of a limit set.- 6 Varia.- 6.1 Complexity growth.- 6.2 Expansive systems.- 6.3 Pseudo-orbits and pseudoleaves.- 6.4 Generic leaves.