| Chapter 1 The Mean and Pointwise Ergodic Theorems 1.1 The mean ergodic theorem 1.2 The pointwise ergodic theorem 1.2.1 Birkhoff's ergodic theorem through the maximal inequality 1.2.2 Birkhoff's ergodic theorem with no maximal inequality 1.2.3 Maximal inequalities, dominated ergodic theorem and transference 1.2.4 The pointwise ergodic theorem through a variational inequality Chapter 2 Wiener Wintner Pointwise Ergodic Theorems 2.1 Introduction 2.2 Ergodic transformations, Kronecker factors, spectral measures 2.3 Wiener Wintner theorem through the affinity of measures 2.3.1 Preliminaries on sequences having a correlation and the affinity 2.3.2 First proof of the Wiener Wintner ergodic theorem 2.4 Wiener Wintner theorem through a simple inequality 2.4.1 A simple variant of Van der Corput's inequality 2.4.2 J. Bourgain's uniform Wiener Wintner ergodic theorem 2.5 Wiener Wintner theorem through disjointness 2.5.1 Disjointness and generic points 2.5.2 The third proof 2.6 Topological Wiener Wintner ergodic theorem 2.6.1 Topological dynamical systems 2.6.2 Wiener Wintner results for uniquely ergodic systems 2.7 Remarks and questions 2.7.1 Ergodic decomposition 2.7.2 Comments 2.7.3 Remarks Chapter 3 Universal Weights for Dynamical Systems 3.1 Introduction 3.2 Independent variables as universal weights for the pointwise ergodic theorem 3.2.1 Independent variables as universal weights for the pointwise convergence in L2 3.2.2 Independent random variables as universal weights for the pointwise convergence in Lp Chapter 4 J. Bourgain~s Return Times Theorem 4.1 Introduction 4.2 Preliminaries 4.3 A proof of the return times theorem 4.3.1 Proof for f x 4.3.2 Proof for f x 4.3.2.1 Continuity of the spectral measure of f 4.3.2.2 The finite range assumption 4.3.2.3 Part I 4.3.2.4 Part II 4.3.2.5 Part III 4.3.2.6 Part IV 4.3.2.7 Part V 4.3.2.8 The contradiction 4.3.2.9 Extending beyond the finite range assumption …… Chapter 5 Extensions of the Return Times Theorem Chapter 6 Speed of Convergence in the Uniform Wiener Wintner Theorem Chapter 7 Weak Wiener Wintner Dynamical Systems Chapter 8 Polynomial Wiener Wintner Ergodic Theorem Chapter 9 Extension to More General Operators Bibliography Index |
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