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 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. Part I focuses on volume preserving homeomorphisms of the unit n-dimensional cube. Parts II and III consider compact manifolds and sigma compact manifolds respectively, describing the work of the authors in extending the celebrated result of Oxtoby and Ulam that for volume homeomorphisms of the unit cube, ergodicity is a typical property.