Typical Dynamics of Volume Preserving Homeomorphisms (Cambridge Tracts in Mathematics)
Cambridge University Press, 3/29/2001
EAN 9780521582872, ISBN10: 0521582873
Hardcover, 240 pages, 22.9 x 15.2 x 1.8 cm
Language: English
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.
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.
'An interesting piece of research for the specialist.' Mathematika
'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.' Monatshefe für Mathematik