Typical Dynamics of Volume Preserving Homeomorphisms

Erscheinungsjahr 2010
ISBN: 978-0-521-58287-2
Verlag: Cambridge University Press

Buch, Englisch, Band 139, 236 Seiten, Format (B × H): 157 mm x 235 mm, Gewicht: 546 g

Reihe: Cambridge Tracts in Mathematics

This book provides a self-contained introduction to typical properties of volume preserving homeomorphisms, examples of which include transitivity, chaos and ergodicity. A key notion 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 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 which would be suitable for an undergraduate analysis seminar or a graduate lecture course. Parts II and III consider compact manifolds and sigma compact manifolds respectively. Much of this work describes the work of the two authors 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.

Autoren/Hrsg.

Alpern, Steve

Alpern, S. R.

Prasad, V. S.

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Lineare und multilineare Algebra, Matrizentheorie
• Mathematik | Informatik Mathematik Algebra Homologische Algebra

Inhaltsverzeichnis

Part I. Volume Preserving Homomorphisms of the Cube: 1. Introduction to Parts 1, 2 (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, ?]; 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 3; 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; Other properties typical in M[X, µ]; Appendix 1. Multiple Rokhlin towers and conjugacy approximation; Appendix 2. Homeomorphic measures.