Dynamical Systems and Ergodic Theory

This book is essentially a self-contained introduction to topological dynamics and ergodic theory. It is divided into a number of relatively short chapters with the intention that each may be used as a component of a lecture course tailored to the particular audience. Parts of the book are suitable for a final year undergraduate course or for a master's level course. A number of applications are given, principally to number theory and arithmetic progressions (through van der Waerden's theorem and Szemerdi's theorem).

Introduction

ix

(2)

Preliminaries

xi

1. Conventions

xi

(1)

2. Notation

xi

(1)

3. Pre-requisities in point set topology (Chapters 1-6)

xi

(1)

4. Pre-requisities in measure theory (Chapters 7-12)

xii

(1)

5. Subadditive sequences

xiii

(1)

6. References

xiii

Chapter 1. Examples and basic properties

1

(10)

1.1. Examples

1

(1)

1.2. Transitivity

2

(2)

1.3. Other characterizations of transitivity

4

(1)

1.4. Transitivity for subshifts of finite type

5

(1)

1.5. Minimality and the Birkhoff recurrence theorem

6

(2)

1.6. Commuting homeomorphisms

8

(1)

1.7. Comments and references

9

(2)

Chapter 2. An application of recurrence to arithmetic progressions

11

(8)

2.1. Van der Waerden's theorem

11

(1)

2.2. A dynamical proof

12

(3)

2.3. The proofs of Sulemma 2.2.2 and Sublemma 2.2.3

15

(2)

2.4. Comments and references

17

(2)

Chapter 3. Topological entropy

19

(14)

3.1. Definitions

19

(4)

3.2. The Perron-Frobenius theorem and subshifts of finite type

23

(3)

3.3. Other definitions and examples

26

(4)

3.4. Conjugacy

30

(2)

3.5. Comments and references

32

(1)

Chapter 4. Interval maps

33

(14)

4.1. Fixed points and periodic points

33

(4)

4.2. Topological entropy of interval maps

37

(2)

4.3. Markov maps

39

(5)

4.4. Comments and references

44

(3)

Chapter 5. Hyperbolic toral automorphisms

47

(10)

5.1. Definitions

47

(2)

5.2. Entropy for Hyperbolic Toral Automorphisms

49

(3)

5.3. Shadowing and semi-conjugacy

52

(3)

5.4. Comments and references

55

(2)

Chapter 6. Rotation numbers

57

(8)

6.1. Homeomorphisms of the circle and rotation numbers

57

(3)

6.2. Denjoy's theorem

60

(4)

6.3. Comments and references

64

(1)

Chapter 7. Invariant measures

65

(8)

7.1. Definitions and characterization of invariant measures

65

(1)

7.2. Borel sigma-algebras for compact metric spaces

65

(2)

7.3. Examples of invariant measures

67

(2)

7.4. Invariant measures for other actions

69

(2)

7.5. Comments and references

71

(2)

Chapter 8. Measure theoretic entropy

73

(18)

8.1. Partitions and conditional expectations

73

(3)

8.2. The entropy of a partition

76

(3)

8.3. The entropy of a transformation

79

(3)

8.4. The increasing martingale theorem

82

(2)

8.5. Entropy and sigma algebras

84

(2)

8.6. Conditional entropy

86

(1)

8.7. Proofs of Lemma 8.7 and Lemma 8.8

87

(1)

8.8. Isomorphism

88

(1)

8.9. Comments and references

89

(2)

Chapter 9. Ergodic measures

91

(8)

9.1. Definitions and characterization of ergodic measures

91

(1)

9.2. Poincare recurrence and Kac's theorem

91

(2)

9.3. Existence of ergodic measures

93

(1)

9.4. Some basic constructions in ergodic theory

94

(3)

9.4.1. Skew products

95

(1)

9.4.2. Induced transformations and Rohlin towers

95

(1)

9.4.3. Natural extensions

96

(1)

9.5. Comments and references

97

(2)

Chapter 10. Ergodic theorems

99

(14)

10.1. The Von Neumann ergodic theorem

99

(3)

10.2. The Birkhoff theorem (for ergodic measures)

102

(4)

10.3. Applications of the ergodic theorems

106

(5)

10.4. The Birkhoff theorem (for invariant measures)

111

(1)

10.5. Comments and references

112

(1)

Chapter 11. Mixing Properties

113

(12)

11.1. Weak mixing

113

(1)

11.2. A density one convergence characterization of weak mixing

114

(2)

11.3. A generalization of the Von Neumann ergodic theorem

116

(2)

11.4. The spectral viewpoint

118

(2)

11.5. Spectral characterization of weak mixing

120

(2)

11.6. Strong mixing

122

(1)

11.7. Comments and reference

123

(2)

Chapter 12. Statistical properties in ergodic theory

125

(14)

12.1. Exact endomorphisms

125

(1)

12.2. Statistical properties of piecewise expanding Markov maps

126

(7)

12.3. Rohlin's entropy formula

133

(1)

12.4. The Shannon-McMillan-Brieman theorem

134

(3)

12.5. Comments and references

137

(2)

Chapter 13. Fixed points for homeomorphisms of the annulus

139

(8)

13.1. Fixed points for the annulus

139

(5)

13.2. Outline proof of Brouwer's theorem

144

(2)

13.3. Comments and references

146

(1)

Chapter 14. The variational principle

147

(6)

14.1. The variational principle for entropy

147

(1)

14.2. The proof of the variational principle

147

(5)

14.3. Comments and reference

152

(1)

Chapter 15. Invariant measures for commuting transformations

153

(8)

15.1. Furstenberg's conjecture and Rudolph's theorem

153

(1)

15.2. The proof of Rudolph's theorem

153

(6)

15.3. Comments and references

159

(2)

Chapter 16. Multiple recurrence and Szemeredi's theorem

161

(16)

16.1. Szemeredi's theorem on arithmetic progressions

161

(1)

16.2. An ergodic proof of Szemeredi's theorem

162

(1)

16.3. The proof of Theorem 16.2

163

(3)

16.3.1. (UMR) for weak-mixing systems, weak-mixing extensions and compact systems

163

(2)

16.3.2. The non-weak-mixing case

165

(1)

16.3.3. (UMR) for compact extensions

165

(1)

16.3.4. The last step

165

(1)

16.4. Appendix to section 16.3

166

(10)

16.4.1. The proofs of Propositions 16.3 and 16.4

166

(5)

16.4.2. The proof of Proposition 16.5

171

(1)

16.4.3. The proof of Proposition 16.6

171

(1)

16.4.4. The proof of Proposition 16.7

172

(1)

16.4.5. The proof of Proposition 16.8

173

(2)

16.4.6. The proof of Proposition 16.9

175

(1)

16.5. Comments and references

176

(1)

Index

177

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Dynamical Systems and Ergodic Theory

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