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Typical Dynamics of Volume Preserving Homeomorphisms

Part of Cambridge Tracts in Mathematics

  • Date Published: February 2011
  • availability: Available
  • format: Paperback
  • isbn: 9780521172431


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About the Authors
  • 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.

    • Presents a self-contained introduction to this area
    • Parts of the book are suitable for graduate courses
    • Authors are widely respected for their research over twenty years
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    Reviews & endorsements

    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 für Mathematik

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    Product details

    • Date Published: February 2011
    • format: Paperback
    • isbn: 9780521172431
    • length: 238 pages
    • dimensions: 229 x 152 x 14 mm
    • weight: 0.35kg
    • availability: Available
  • Table of Contents

    Historical Preface
    General outline
    Part I. Volume Preserving Homomorphisms of the Cube:
    1. Introduction to Parts I and II (compact manifolds)
    2. Measure preserving homeomorphisms
    3. Discrete approximations
    4. Transitive homeomorphisms of In and Rn
    5. Fixed points and area preservation
    6. Measure preserving Lusin theorem
    7. Ergodic homeomorphisms
    8. Uniform approximation in G[In, λ] and generic properties in Μ[In, λ]
    Part II. Measure Preserving Homeomorphisms of a Compact Manifold:
    9. Measures on compact manifolds
    10. Dynamics on compact manifolds
    Part III. Measure Preserving Homeomorphisms of a Noncompact Manifold:
    11. Introduction to Part III
    12. Ergodic volume preserving homeomorphisms of Rn
    13. Manifolds where ergodic is not generic
    14. Noncompact manifolds and ends
    15. Ergodic homeomorphisms: the results
    16. Ergodic homeomorphisms: proof
    17. Other properties typical in M[X, μ]
    Appendix 1. Multiple Rokhlin towers and conjugacy approximation
    Appendix 2. Homeomorphic measures

  • Authors

    Steve Alpern, London School of Economics and Political Science

    V. S. Prasad, University of Massachusetts, Lowell

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