Typical Dynamics of Volume Preserving Homeomorphisms

AvSteve Alpern,V. S. Prasad

Häftad, Engelska, 2011

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Beskrivning

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.

Produktinformation

Utgivningsdatum:2011-02-17
Mått:152 x 229 x 14 mm
Vikt:350 g
Format:Häftad
Språk:Engelska
Serie:Cambridge Tracts in Mathematics
Antal sidor:238
Förlag:Cambridge University Press
ISBN:9780521172431

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Review of the hardback: 'An interesting piece of research for the specialist.' Mathematika

Innehållsförteckning

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; Bibliography; Index.