Typical dynamics of volume preserving homeomorphisms

Steve Alpern, V. S. Prasad0521582873, 9780521582872, 9780511039836

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.

---

Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Dedication……Page 7
Acknowledgements……Page 8
Contents……Page 9
Historical Preface……Page 13
General Outline……Page 18
Part I Volume Preserving Homeomorphisms of the Cube……Page 23
1.2 Automorphisms of a Measure Space……Page 25
1.3 Main Results for Compact Manifolds……Page 26
2.1 The Spaces M, H, G……Page 29
2.2 Extending a Finite Map……Page 31
3.1 Introduction……Page 35
3.2 Dyadic Permutations……Page 36
3.3 Cyclic Dyadic Permutations……Page 38
3.4 Rotationless Dyadic Permutations……Page 40
4.1 Transitive Homeomorphisms……Page 44
4.2 A Transitive Homeomorphism of I……Page 45
4.3 A Transitive Homeomorphism of R……Page 46
4.4 Topological Weak Mixing……Page 47
4.5 A Chaotic Homeomorphism of I……Page 49
4.6 Periodic Approximations……Page 51
5.1 Introduction……Page 53
5.2 The Plane Translation Theorem……Page 54
5.3 The Open Square……Page 55
5.4 The Torus……Page 57
5.5 The Annulus……Page 58
6.1 Introduction……Page 60
6.2 Approximation Techniques……Page 63
6.3 Proof of Theorem 6.2(i)……Page 67
7.1 Introduction……Page 70
7.2 A Classical Proof of Generic Ergodicity……Page 72
8.1 Introduction……Page 75
8.2 Rokhlin Towers and Stochastic Matrices……Page 77
Part II Measure Preserving Homeomorphisms of a Compact Manifold……Page 81
9.2 General Measures on the Cube……Page 83
9.3 Manifolds……Page 86
9.4 Measures on Compact Manifolds……Page 88
9.5 Typical Properties in M[X, µ]……Page 91
10.2 Genericity Results for Manifolds……Page 93
10.3 Applications to Fixed Point Theory……Page 97
Part III Measure Preserving Homeomorphisms of a Noncompact Manifold……Page 101
11.2 Topologies on G[X, µ] and M[X, µ]: Noncompact Case……Page 103
11.3 Main Results for Sigma Compact Manifolds……Page 106
11.4 Outline of Part III……Page 108
12.1 Introduction……Page 111
12.2 Homeomorphisms of R with Invariant Cubes……Page 112
12.3 Generic Ergodicity in M[R, Lambda]……Page 115
12.4 Other Typical Properties in M[R, Lambda]……Page 116
13.2 Two Examples……Page 120
13.3 Ends of a Manifold: Informal Introduction……Page 124
13.6 The Flip on Manhattan……Page 126
13.7 Shear Map on the Strip……Page 127
14.2 End Compactification……Page 128
14.3 Examples of End Compactifications……Page 129
14.4 Algebra Q of Clopen Sets……Page 130
14.5 Measures on Ends……Page 131
14.6 Compact Separating Sets……Page 134
14.7 End Preserving Lusin Theorem……Page 135
14.8 Induced Homeomorphism h……Page 137
14.9 The Charge Induced by a Homeomorphism……Page 143
14.10 h-moving Separating Sets……Page 148
14.11 End Conditions for Homeomorphic Measures……Page 150
15.1 Introduction……Page 152
15.2 Consequences of Theorem 15.1……Page 154
16.1 Introduction……Page 159
16.2 Outline of Proofs of Theorems 15.1 and 15.2……Page 160
16.3 Proof of Theorem 15.1: Strip Manifold……Page 162
16.4 Proofs of Theorems 15.1 and 15.2: General Case……Page 165
17.1 A General Existence Result……Page 176
17.2 Proof of Theorem 17.1……Page 177
17.3 Weak Mixing End Homeomorphisms……Page 179
17.4 Maximal Chaos on Noncompact Manifolds……Page 180
A1.1 Introduction……Page 182
A1.2 Skyscraper Constructions……Page 183
A1.3 Multiple Tower Rokhlin Theorem……Page 188
A1.4 Pointwise Conjugacy Approximation……Page 196
A1.5 Specified Transition Probabilities……Page 199
A1.6 Setwise Conjugacy Approximation……Page 201
A1.7 Infinite Measure Constructions……Page 205
A2.1 Introduction……Page 210
A2.2 Homeomorphic Measures on the Cube……Page 211
A2.3 Homeomorphic Measures on Compact Manifolds……Page 217
A2.4 Homeomorphic Measures on Noncompact Manifolds……Page 218
A2.5 Proof of the Berlanga–Epstein Theorem……Page 220
Bibliography……Page 227
Index……Page 235