Some topics in ergodic theory

Contents Chapter 1. Introduction 5 Chapter 2. Measure-preserving transformations and other basic notions of ergodic theory for deterministic systems 11 1. Example: circle rotations 11 2. Measure-preserving transformations. Formal definitions 13 3. Poincaré's recurrence theorem 15 4. Stochastic processes: basic notions 17 5. Interpretation of measure-preserving maps via stationary processes 19 6. Ergodicity 20 Chapter 3. Ergodic theorems for measure-preserving transformations 25 1. Von Neumann's ergodic theorem in L 2. 25 2. Birkhoff's pointwise ergodic theorem 28 3. Kingman's subadditive ergodic theorem 32 Chapter 4. Invariant measures 37 1. Existence of invariant measures 37 2. Structure of the set of invariant measures. 38 3. Absolutely continuous invariant measures 45 Chapter 5. Random dynamics 49 1. Stability in stochastic dynamics 49 2. Markov processes and random transformations 52 3. Invariant measures 57 4. Ergodicity for Markov processes and systems of i.i.d. transformations 64 Chapter 6. Invariant distributions for Markov chains in discrete spaces 67 1. Perron–Frobenius Theorem 67 2. Hilbert projective metric approach. 69 3. Another proof of ergodic theorem for Markov chain 71 4. Entropy approach 72 5. Coupling 74 6. Classification of states of a Markov chain 76 3 4 CONTENTS Chapter 7. Invariant distributions for Markov processes in general spaces 81 1. Existence via Krylov–Bogolyubov approach 81 2. Doeblin condition 82 3. Harris positive recurrence condition 84 Bibliography 87 CHAPTER 1 Introduction There is no universally accepted concise definition of ergodic theory. I am going to adopt the view that its main purpose is to study statistical patterns in deterministic or random dynamical systems and, specifically, how these patterns depend on the initial state of the system. Let me first try to explain this without going to technical details. Suppose that we have a system X evolving in a phase space X. This means that at each time t the state of the system denoted by X t belongs to X. The system may be rather complex, e.g., X can be a turbulent flow of particles in a fluid or gas and at any given time the description of the state of the flow should involve positions and velocities of all these particles. All these details of the flow at any given time have to be encoded as an element of X. Studying such a system experimentally may be a difficult task because if the behavior of the system is rich, then tracking all the details of the …