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Typical Dynamics of Volume Preserving Homeomorphisms : Cambridge Tracts in Mathematics - Steve Alpern

Typical Dynamics of Volume Preserving Homeomorphisms

Cambridge Tracts in Mathematics


Published: 17th February 2011
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This 2000 book provides a self-contained introduction to typical properties of homeomorphisms. Examples of properties of homeomorphisms considered include transitivity, chaos and ergodicity. A key idea here is the interrelation between typical properties of volume preserving homeomorphisms and typical properties of volume preserving bijections of the underlying measure space. The authors make the first part of this book very concrete by considering volume preserving homeomorphisms of the unit n-dimensional cube, and they go on to prove fixed point theorems (Conley-Zehnder- Franks). This is done in a number of short self-contained chapters which would be suitable for an undergraduate analysis seminar or a graduate lecture course. Much of this work describes the work of the two authors, over the last twenty years, in extending to different settings and properties, the celebrated result of Oxtoby and Ulam that for volume homeomorphisms of the unit cube, ergodicity is a typical property.

Review of the hardback: 'An interesting piece of research for the specialist.' Mathematika
Review of the hardback: 'The authors of this book are undoubtedly the experts of generic properties of measure preserving homeomorphisms of compact and locally compact manifolds, continuing and extending ground-breaking early work by J. C. Oxtoby and S. M. Ulam. The book is very well and carefully written and is an invaluable reference for anybody working on the interface between topological dymanics and ergodic theory.' Monatshefte fur Mathematik

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
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780521172431
ISBN-10: 0521172438
Series: Cambridge Tracts in Mathematics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 238
Published: 17th February 2011
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.9 x 15.2  x 1.4
Weight (kg): 0.35