Typical Dynamics of Volume Preserving Homeomorphisms

by Steve Alpern , V. S. Prasad

• ISBN13: 9780521172431
• ISBN10: 0521172438
• Format: Paperback
• Copyright: 2011-02-17
• Publisher: Cambridge University Press

Summary

This book provides a self-contained introduction to typical properties of volume preserving homeomorphisms, examples of which include transitivity, chaos and ergodicity. The authors make the first part of the book very concrete by focusing on volume preserving homeomorphisms of the unit n-dimensional cube. They also prove fixed point theorems (Conley-Zehnder-Franks). This is done in a number of short self-contained chapters that would be suitable for an undergraduate analysis seminar or a graduate lecture course. Parts Two and Three consider compact manifolds and sigma compact manifolds respectively, describing the work of the two authors in extending the celebrated result of Oxtoby and Ulam that for volume homeomorphisms of the unit cube, ergodicity is a typical property.

Table of Contents

Historical Preface

General outline

Volume Preserving Homomorphisms of the Cube

Introduction to Parts I and II (compact manifolds)

Measure preserving homeomorphisms

Discrete approximations

Transitive homeomorphisms of In and Rn

Fixed points and area preservation

Measure preserving Lusin theorem

Ergodic homeomorphisms

Uniform approximation in G[In, Δ] and generic properties in Σ[In, Δ]

Measure Preserving Homeomorphisms of a Compact Manifold

Measures on compact manifolds

Dynamics on compact manifolds

Measure Preserving Homeomorphisms of a Noncompact Manifold

Introduction to Part III

Ergodic volume preserving homeomorphisms of Rn

Manifolds where ergodic is not generic

Noncompact manifolds and ends

Ergodic homeomorphisms: the results

Ergodic homeomorphisms: proof

Other properties typical in M[X, μ]

Multiple Rokhlin towers and conjugacy approximation

Homeomorphic measures

Bibliography

Index

Table of Contents provided by Publisher. All Rights Reserved.