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Dynamics of Foliations, Groups and Pseudogroups de Pawel Walczak

Dynamics of Foliations, Groups and Pseudogroups: Monografie Matematyczne, cartea 64

Autor Pawel Walczak
en Limba Engleză Hardback – 23 apr 2004
Foliations, groups and pseudogroups are objects which are closely related via the notion of holonomy. In the 1980s they became considered as general dynamical systems. This book deals with their dynamics. Since "dynamics” is a very extensive term, we focus on some of its aspects only. Roughly speaking, we concentrate on notions and results related to different ways of measuring complexity of the systems under consideration. More precisely, we deal with different types of growth, entropies and dimensions of limiting objects. Invented in the 1980s (by E. Ghys, R. Langevin and the author) geometric entropy of a foliation is the principal object of interest among all of them.
Throughout the book, the reader will find a good number of inspirating problems related to the topics covered.
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Specificații

ISBN-13: 9783764370916
ISBN-10: 3764370912
Pagini: 228
Ilustrații: XI, 228 p.
Dimensiuni: 155 x 235 x 20 mm
Greutate: 0.67 kg
Ediția:2004
Editura: Birkhäuser Basel
Colecția Birkhäuser
Seria Monografie Matematyczne

Locul publicării:Basel, Switzerland

Public țintă

Research

Cuprins

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.

Recenzii

"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
 

Caracteristici

Existing books on foliations deal mainly with either topology or Riemannian geometry of them; books on dynamics deal either with classical systems (single transformations or flows) or with groups. Here, dynamics of foliations (almost absent in the existing books) interplays with that of groups and pseudogroups