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  1. Home
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  3. Typical Dynamics of Volume Preserving Homeomorphisms
  • Cited by 13
Publisher:
Cambridge University Press
Online publication date:
August 2009
Print publication year:
2001
Online ISBN:
9780511543180
DOI:
https://doi.org/10.1017/CBO9780511543180
Subjects:
Mathematics (general), Mathematics, Differential and Integral Equations, Dynamical Systems and Control Theory
Series:
Cambridge Tracts in Mathematics (139)
Information

Book description

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.

Reviews

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

Source: Mathematika

Review of the hardback:'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.'

Source: Monatshefte für Mathematik

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