Typical Dynamics of Volume Preserving Homeomorphisms

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.

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, ?] and generic properties in ?[In, ?]; 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.

This 2000 book provides a self-contained introduction to typical properties of volume preserving homeomorphisms. Stress is given to the interrelation between typical properties of volume preserving homeomorphisms and typical properties of volume preserving bijections of the underlying measure space. An excellent introduction for newcomers, and an indispensable resource for experts.