Let $f \colon X \to X$ be a topological dynamical system on a compact metric space. We denote by $\mathcal{M}(f) \subseteq \mathcal{M}(X)$ the Borel probability measures on $X$ that are invariant under $f$ (i.e. measures for which $f$ becomes a measure-preserving dynamical system).

In this lecture we use the Markov-Kakutani Fixed Point Theorem to show that the space $\mathcal{M}(f)$ is a non-empty compact convex subset of $\mathcal{M}(X)$.

Next, let $\mathcal{E}(f) \subseteq \mathcal{M}(f)$ denote those measures for which $f$ is additionally an ergodic measure-preserving dynamical system.

We prove that the elements of $\mathcal{E}(f)$ are precisely the extremal points of $\mathcal{M}(f)$. We conclude by proving that $\mathcal{E}(f)$ is also non-empty[1].

1. In finite dimensions, it is geometrically obvious that the extremal points of a convex set are non-empty. This is still true in infinite dimensions, but it is much harder to prove (this is called the Krein-Milman Theorem). Rather than quote this theorem however we provide a direct proof. ↩︎