Book contents
- Frontmatter
- Contents
- Introduction
- Preliminaries
- Chapter 1 Examples and basic properties
- Chapter 2 An application of recurrence to arithmetic progressions
- Chapter 3 Topological entropy
- Chapter 4 Interval maps
- Chapter 5 Hyperbolic toral automorphisms
- Chapter 6 Rotation numbers
- Chapter 7 Invariant measures
- Chapter 8 Measure theoretic entropy
- Chapter 9 Ergodic measures
- Chapter 10 Ergodic theorems
- Chapter 11 Mixing Properties
- Chapter 12 Statistical properties in ergodic theory
- Chapter 13 Fixed points for homeomorphisms of the annulus
- Chapter 14 The variational principle
- Chapter 15 Invariant measures for commuting transformations
- Chapter 16 Multiple recurrence and Szemeredi's theorem
- Index
Chapter 9 - Ergodic measures
Published online by Cambridge University Press: 05 March 2015
- Frontmatter
- Contents
- Introduction
- Preliminaries
- Chapter 1 Examples and basic properties
- Chapter 2 An application of recurrence to arithmetic progressions
- Chapter 3 Topological entropy
- Chapter 4 Interval maps
- Chapter 5 Hyperbolic toral automorphisms
- Chapter 6 Rotation numbers
- Chapter 7 Invariant measures
- Chapter 8 Measure theoretic entropy
- Chapter 9 Ergodic measures
- Chapter 10 Ergodic theorems
- Chapter 11 Mixing Properties
- Chapter 12 Statistical properties in ergodic theory
- Chapter 13 Fixed points for homeomorphisms of the annulus
- Chapter 14 The variational principle
- Chapter 15 Invariant measures for commuting transformations
- Chapter 16 Multiple recurrence and Szemeredi's theorem
- Index
Summary
In this chapter we shall consider the stronger property of ergodicity for an invariant probability measure μ. This property is more appropriate (amongst other things) for understanding the “long term” average behaviour of a transformation.
Definitions and characterization of ergodic measures
Definition. Given a probability space (X, B, μ), a transformation T : X → X is called ergodic if for every set B ∈ B with T−1B = B we have that either μ(B) = 0 or μ(B) = 1.
Alternatively we say that μ is T-ergodic.
The following lemma gives a simple characterization in terms of functions.
Lemma 9.1. T is ergodic with respect to μ iff whenever f ∈ L1(X, B, μ) satisfies f = f ∘ T then f is a constant function.
Proof. This is an easy observation using indicator functions.
Poincaré recurrence and Kac's theorem
We begin with one of the most fundamental results in ergodic theory.
Theorem 9.2 (Poincaré recurrence theorem). Let T : X → X be a measurable transformation on a probability space (X, B, μ). Let A ∈ B have μ(A) > 0; then for almost points x ∈ A the orbit {Tnx}n ≥ 0 returns to A infinitely often.
Proof. Let F = {x ∈ A : Tnx ∉ A, ∀n ≥ 1}, then it suffices to show that μ(F) = 0.
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- Dynamical Systems and Ergodic Theory , pp. 91 - 98Publisher: Cambridge University PressPrint publication year: 1998