Ergodic Theorems

✍ Scribed by Ulrich Krengel; Antoine Brunel

• Publisher: De Gruyter
• Year: 1985
• Tongue: English
• Leaves: 368
• Series: De Gruyter Studies in Mathematics; 6
• Category: Library

✦ Table of Contents

Chapter 1: Measure preserving and null preserving point mappings
§ 1.1 Von Neumann’s mean ergodic theorem, ergodicity
§ 1.2 Birkhoff’s ergodic theorem
§ 1.3 Recurrence
§ 1.4 Shift transformations and stationary processes
§ 1.5 Kingman’s subadditive ergodic theorem and the multiplicative ergodic theorem of Oseledec
§ 1.6 Relatives of the maximal ergodic theorem
§ 1.7 Some general tools and principles
Chapter 2: Mean ergodic theory
§ 2.1 The mean ergodic theorem
§ 2.2 Uniform convergence
§ 2.3 Weak mixing, continuous spectrum and multiple recurrence
§ 2.4 The splitting theorem of Jacobs-Deleeuw-Glicksberg
Chapter 3: Positive contractions in L1
§ 3.1 The Hopf decomposition
§ 3.2 The Chacon-Ornstein theorem
§ 3.3 Brunel’s lemma and the identification of the limit
§ 3.4 Existence of finite invariant measures
§ 3.5 The subadditive ergodic theorem for positive contractions in L1
§ 3.6 An example with divergence of Cesàro averages
§ 3.7 More on the filling scheme
Chapter 4: Extensions of the L1-theory
§ 4.1 Non positive contractions in L1
§ 4.2 Vector valued ergodic theorems
§ 4.3 Power bounded operators and harmonic functions
Chapter 5: Operators in C(K) and in Lp, (1<p<∞)
§ 5.1 Markov operators in C(K)
§ 5.2 Contractions in Lp, (1 <p < ∞)
Chapter 6: Pointwise ergodic theorems for multiparameter and amenable semigroups
§ 6.1 Unrestricted convergence for averages over d-dimensional intervals
§ 6.2 Multiparameter additive and subadditive processes
§ 6.3 Multiparameter semigroups of L1-contractions
§ 6.4 Amenable semigroups
Chapter 7: Local ergodic theorems and differentiation
§ 7.1 Positive 1-parameter semigroups
§ 7.2 Local ergodic theorems for multiparameter and non positive semigroups, and for vector valued functions
Chapter 8: Subsequences and generalized means
§ 8.1 Strong convergence and mixing
§ 8.2 Pointwise convergence
Chapter 9: Special topics
§ 9.1 Ergodic theorems in von Neumann algebras
§ 9.2 Entropy and information
§ 9.3 Nonlinear nonexpansive mappings
§ 9.4 Miscellanea
Supplement: Harris Processes, Special Functions, Zero-Two-Law (by Antoine Brunei)
Bibliography
Notation
Index