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.
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Editorial Reviews
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Review of the hardback: 'An interesting piece of research for the specialist.' 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.' Monatshefte für Mathematik
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.' Monatshefte für Mathematik
Book Description
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.
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Inside This Book
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First Sentence:
Two of the principal analytic structures that may be put on a set X are measure and topology. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
end homeomorphism, homeomorphic measures, dyadic permutations, ergodic homeomorphisms, mean rotation zero, generic ergodicity, volume preserving homeomorphisms, chaotic homeomorphisms, strip manifold, ergodic approximation, maximal chaos, topological weak mixing, tower theorem, measure preserving homeomorphisms, invariant cubes, transitive homeomorphism, bad cubes, torus homeomorphisms, area preserving homeomorphism, measure theoretic properties, measure theoretic property, ergodic automorphism, sigma compact, last geometric theorem, dyadic cubes Key Phrases - Capitalized Phrases (CAPs): (learn more)
Baire Category Theorem, End Preserving Lusin Theorem, Proof Let, Plane Translation Theorem, Setwise Conjugacy Approximation Theorem, Annulus Theorem, Proof First, Proof Suppose, Proofs of Theorems, Franks's Theorem, Poincaré Recurrence Theorem, Brouwer's Theorem, Finite Map, Other Properties Typical, Peter Lax, Rokhlin's Tower Theorem New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Two of the principal analytic structures that may be put on a set X are measure and topology. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
end homeomorphism, homeomorphic measures, dyadic permutations, ergodic homeomorphisms, mean rotation zero, generic ergodicity, volume preserving homeomorphisms, chaotic homeomorphisms, strip manifold, ergodic approximation, maximal chaos, topological weak mixing, tower theorem, measure preserving homeomorphisms, invariant cubes, transitive homeomorphism, bad cubes, torus homeomorphisms, area preserving homeomorphism, measure theoretic properties, measure theoretic property, ergodic automorphism, sigma compact, last geometric theorem, dyadic cubes Key Phrases - Capitalized Phrases (CAPs): (learn more)
Baire Category Theorem, End Preserving Lusin Theorem, Proof Let, Plane Translation Theorem, Setwise Conjugacy Approximation Theorem, Annulus Theorem, Proof First, Proof Suppose, Proofs of Theorems, Franks's Theorem, Poincaré Recurrence Theorem, Brouwer's Theorem, Finite Map, Other Properties Typical, Peter Lax, Rokhlin's Tower Theorem New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 4
books:
- Measure Theory (Graduate Texts in Mathematics) (v. 18) by Paul R. Halmos in Back Matter
- Adventures of a Mathematician by S. M. Ulam in Front Matter
- Discrete Mathematics by Wiley in Back Matter
- Contributions to the Theory of Games, Vol. III (Annals of Mathematics Studies) by Melvin Dresher in Back Matter
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