Dynamics of Foliations, Groups and Pseudogroups

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.- 3.2 Entropy of pseudogroups.- 3.3 Geometric entropy of foliations.- 3.4 Relating various entropies.- 3.5 Examples and constructions.- 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.

"The classical theory of dynamical systems has been greatly generalized to the rich context of foliations and actions of groups and pseudogroups on spaces. The book under review expounds on this theory in considerable detail....

Much of the material in this book is pertinent to applied mathematics.... Experts in control systems also recognize foliation theory as a cognate subject of some interest.... With the phenomenal interaction between pure and applied mathematics over recent decades, this book should be of considerable interest to many application-oriented mathematicians."

-SIAM Book Reviews