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Table of Contents
Random Walks on Reductive Groups; Contents; Chapter 1: Introduction; 1.1 What Is This Book About?; 1.2 When Did This Topic Emerge?; 1.3 Is This Topic Related to Sums of Random Numbers?; 1.4 What Classical Results Should I Know?; 1.5 Can You Show Me Some Nice Sample Results from This Topic?; 1.6 How Does One Prove These Nice Results?; 1.7 Can You Answer Your Own Questions Now?; 1.8 Where Can I Find These Answers in This Book?; 1.9 Why Is This Book Less Simple than These Samples?; 1.10 Can You State These More General Limit Theorems?; 1.11 Are the Proofs as Simple as for the Simple Samples?
1.12 Why Is the Iwasawa Cocycle so Important to You?1.13 I Am Allergic to Local Fields. Is It Safe to Open This Book?; 1.14 Why Are There so Many Chapters in This Book?; 1.15 Whom Do You Thank?; Part I: The Law of Large Numbers; Chapter 2: Stationary Measures; 2.1 Markov Operators; 2.1.1 Markov Chains on Standard Borel Spaces; 2.1.2 Measure-Preserving Markov Operators; 2.1.3 Ergodicity of Markov Operators; 2.2 Ergodicity and the Forward Dynamical System; 2.3 Markov-Feller Operators; 2.4 Stationary Measures and the Forward Dynamical System
2.5 The Limit Measures and the Backward Dynamical System2.6 The Two-Sided Fibered Dynamical System; 2.7 Proximal Stationary Measures; Chapter 3: The Law of Large Numbers; 3.1 Birkhoff Averages for Functions on GxX; 3.2 Breiman's Law of Large Numbers; 3.3 The Law of Large Numbers for Cocycles; 3.3.1 Random Walks on X; 3.3.2 Cocycles; 3.3.3 The Law of Large Cocycles; 3.3.4 The Invariance Property; 3.4 Convergence of the Covariance 2-Tensors; 3.4.1 Special Cocycles; 3.4.2 The Covariance Tensor; 3.5 Divergence of Birkhoff Sums; Chapter 4: Linear Random Walks; 4.1 Linear Groups
4.2 Stationary Measures on P(V) for V Strongly Irreducible4.3 Virtually Invariant Subspaces; 4.4 Stationary Measures on P(V); 4.5 Norms of Vectors and Norms of Matrices; 4.6 The Law of Large Numbers on P(V); 4.7 Positivity of the First Lyapunov Exponent; 4.8 Proximal and Non-proximal Representations; Chapter 5: Finite Index Subsemigroups; 5.1 The Expected Birkhoff Sum at the First Return Time; 5.2 The First Return in a Finite Index Subsemigroup; 5.3 Stationary Measures for Finite Extensions; 5.4 Cocycles and Finite Extensions; 5.5 A Simple Example (1); Part II: Reductive Groups
Chapter 6: Loxodromic Elements6.1 Basics on Zariski Topology; 6.2 Zariski Dense Semigroups in SL(d,R); 6.3 Zariski Closure of Semigroups; 6.4 Proximality and Zariski Closure; 6.5 Simultaneous Proximality; 6.6 Loxodromic and Proximal Elements; 6.7 Semisimple Real Lie Groups; 6.7.1 Algebraic Groups and Maximal Compact Subgroups; 6.7.2 Cartan Subspaces and Restricted Roots; 6.7.3 Simple Restricted Roots and Weyl Chambers; 6.7.4 The Cartan Projection; 6.7.5 The Iwasawa Cocycle; 6.7.6 The Jordan Projection; 6.7.7 Example: G=SL(d,R); 6.7.8 Example: G=SO(p,q); 6.8 Representations of G
1.12 Why Is the Iwasawa Cocycle so Important to You?1.13 I Am Allergic to Local Fields. Is It Safe to Open This Book?; 1.14 Why Are There so Many Chapters in This Book?; 1.15 Whom Do You Thank?; Part I: The Law of Large Numbers; Chapter 2: Stationary Measures; 2.1 Markov Operators; 2.1.1 Markov Chains on Standard Borel Spaces; 2.1.2 Measure-Preserving Markov Operators; 2.1.3 Ergodicity of Markov Operators; 2.2 Ergodicity and the Forward Dynamical System; 2.3 Markov-Feller Operators; 2.4 Stationary Measures and the Forward Dynamical System
2.5 The Limit Measures and the Backward Dynamical System2.6 The Two-Sided Fibered Dynamical System; 2.7 Proximal Stationary Measures; Chapter 3: The Law of Large Numbers; 3.1 Birkhoff Averages for Functions on GxX; 3.2 Breiman's Law of Large Numbers; 3.3 The Law of Large Numbers for Cocycles; 3.3.1 Random Walks on X; 3.3.2 Cocycles; 3.3.3 The Law of Large Cocycles; 3.3.4 The Invariance Property; 3.4 Convergence of the Covariance 2-Tensors; 3.4.1 Special Cocycles; 3.4.2 The Covariance Tensor; 3.5 Divergence of Birkhoff Sums; Chapter 4: Linear Random Walks; 4.1 Linear Groups
4.2 Stationary Measures on P(V) for V Strongly Irreducible4.3 Virtually Invariant Subspaces; 4.4 Stationary Measures on P(V); 4.5 Norms of Vectors and Norms of Matrices; 4.6 The Law of Large Numbers on P(V); 4.7 Positivity of the First Lyapunov Exponent; 4.8 Proximal and Non-proximal Representations; Chapter 5: Finite Index Subsemigroups; 5.1 The Expected Birkhoff Sum at the First Return Time; 5.2 The First Return in a Finite Index Subsemigroup; 5.3 Stationary Measures for Finite Extensions; 5.4 Cocycles and Finite Extensions; 5.5 A Simple Example (1); Part II: Reductive Groups
Chapter 6: Loxodromic Elements6.1 Basics on Zariski Topology; 6.2 Zariski Dense Semigroups in SL(d,R); 6.3 Zariski Closure of Semigroups; 6.4 Proximality and Zariski Closure; 6.5 Simultaneous Proximality; 6.6 Loxodromic and Proximal Elements; 6.7 Semisimple Real Lie Groups; 6.7.1 Algebraic Groups and Maximal Compact Subgroups; 6.7.2 Cartan Subspaces and Restricted Roots; 6.7.3 Simple Restricted Roots and Weyl Chambers; 6.7.4 The Cartan Projection; 6.7.5 The Iwasawa Cocycle; 6.7.6 The Jordan Projection; 6.7.7 Example: G=SL(d,R); 6.7.8 Example: G=SO(p,q); 6.8 Representations of G