Pollicott M., Yuri M. Dynamical systems and ergodic theory

Contents......Page 1
Introduction
......Page 5
Preliminaries......Page 7
1.1 Examples
......Page 10
1.2 Transitivity......Page 11
1.3 Other characterizations of transitivity
......Page 13
1.4 Transitivity for subshifts of finite type
......Page 14
1.5 Minimality and the Birkhoff recurrence theorem
......Page 15
1.6 Commuting homeomorphisms
......Page 17
1.7 Comments and references
......Page 18
2.1 Van der Waerden's theorem
......Page 19
2.2 A dynamical proof
......Page 20
2.3 The proofs of Sublemma 2.2.2 and Sublemma 2.2.3
......Page 23
2.4 Comments and references
......Page 25
3.1 Defininitions
......Page 26
3.2 The Perron-Frobenius theorem and subshifts of finite type......Page 30
3.3. Other definitions and examples......Page 33
3.4 Conjugacy......Page 37
3.5 Comments and references
......Page 39
4.1 Fixed points and periodic points......Page 40
4.2 Topological entropy of interval maps
......Page 44
4.3 Markov maps......Page 46
4.4 Comments and references
......Page 52
5.1 Definitions
......Page 53
5.2 Entropy for hyperbolic toral automorphisms
......Page 55
5.3 Shadowing and semi-conjugacy
......Page 58
5.4 Comments and references
......Page 61
6.1 Homeomorphisms of the circle and rotation numbers
......Page 62
6.2
Denjoy's theorem......Page 65
6.3 Comments and references
......Page 69
7.2 Borel sigma-algebras for compact metric spaces
......Page 70
7.3 Examples of invariant measures
......Page 72
7.4 Invariant measures for other actions
......Page 74
7.5 Comments and references
......Page 76
8.1 Partitions and conditional expectations......Page 77
8.2 The entropy of a partition......Page 80
8.3 The entropy of a transformation......Page 83
8.4 The increasing martingale theorem
......Page 86
8.5 Entropy and sigma-algebras
......Page 88
8.6 Conditional entropy
......Page 90
8.7 Proofs of Lemma 8.7 and Lemma 8.8
......Page 91
8.8 Isomorphism
......Page 92
8.9 Comments and references
......Page 93
9.2 Poincaré
recurrence and Kac's theorem......Page 94
9.3 Existence of ergodic measures
......Page 96
9.4 Some basic constructions in ergodic theory
......Page 97
9.4.2 Induced transformations and Rohlin towers......Page 98
9.4.3 Natural extensions
......Page 99
9.5
Comments and references......Page 100
10.1 The von Neumann ergodic theorem......Page 101
10.2 The Birkhoff theorem (for ergodic measures)
......Page 104
10.3 Applications of the ergodic theorems
......Page 108
10.4 The Birkhoff theorem (for invariant measures)
......Page 113
10.5 Comments and references
......Page 114
11.1 Weak mixing
......Page 115
11.2 A density one convergence characterization of weak mixing
......Page 116
11.3 A generalization of the von Neumann ergodic theorem......Page 118
11.4 The spectral viewpoint......Page 120
11.5 Spectral characterization of weak mixing......Page 122
11.6 Strong mixing......Page 124
11.7 Comments and references
......Page 125
12.1 Exact endomorphisms
......Page 126
12.2 Statistical properties of piecewise expanding Markov maps
......Page 127
12.3 Rohlin's entropy formula
......Page 134
12.4 The Shannon-McMillan-Brieman theorem......Page 135
12.5 Comments and references
......Page 138
13.1 Fixed points for the annulus
......Page 139
13.2 Outline proof of Brouwer's theorem......Page 144
13.3 Comments and references......Page 146
14.2 The proof of the variational principle
......Page 147
14.3 Comments and references......Page 152
15.2 The proof of Rudolph's theorem......Page 153
15.3 Comments and references
......Page 159
16.1 Szemerdi's theorem on arithmetic progressions......Page 160
16.2 An ergodic proof of Szemerdi's theorem......Page 161
16.3.1 (UMR) for weak-mixing systems, weak-mixing extensions and compact systems......Page 162
16.3.4 The last step......Page 164
16.4.1 The proofs of Propositions 16.3 and 16.4
......Page 165
16.4.3 The proof of Proposition 16.6......Page 170
16.4.4 The proof of Proposition 16.7......Page 171
16.4.5 Proof of Proposition 16.8......Page 172
16.4.6 Proof of Proposition 16.9......Page 174
16.5 Comments and references
......Page 175
Index......Page 176
Errata......Page 179
Discarded chapter......Page 187