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.