Dynamics of Foliations, Groups and Pseudogroups

Author:   Pawel Walczak
Publisher:   Birkhauser Verlag AG
Edition:   2004 ed.
Volume:   64
ISBN:  

9783764370916


Pages:   228
Publication Date:   23 April 2004
Format:   Hardback
Availability:   In Print   Availability explained
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Dynamics of Foliations, Groups and Pseudogroups


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Overview

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.

Full Product Details

Author:   Pawel Walczak
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   2004 ed.
Volume:   64
Dimensions:   Width: 15.50cm , Height: 1.70cm , Length: 23.50cm
Weight:   0.660kg
ISBN:  

9783764370916


ISBN 10:   3764370912
Pages:   228
Publication Date:   23 April 2004
Audience:   College/higher education ,  Professional and scholarly ,  Tertiary & Higher Education ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print   Availability explained
This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us.

Table of Contents

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.

Reviews

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


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


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.... <p>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. <p>a SIAM Book Reviews <p>


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