A Very Brief Introduction to Ergodic Theory

These are expanded notes from four introductory lectures on Ergodic Theory, given at the Minerva summer school Flows on homogeneous spaces at the Technion, Haifa, Israel, in September 2012. 1. Dynamics on a compact metrizable space Given a compact metrizable space X, denote by C(X) the space of continuous functions f : X → C with the uniform norm ‖f‖u = max x∈X |f(x)|. This is a separable Banach space (see Exs 1.2.(a)). Denote by Prob(X) the space of all regular probability measures on the Borel σ-algebra of X. By Riesz representation theorem, the dual C(X)∗ is the space Meas(X) of all finite signed regular Borel measures on X with the total variation norm, and Prob(X) ⊂ Meas(X) is the subset of Λ ∈ C(X)∗ that are positive (Λ(f) ≥ 0 whenever f ≥ 0) and normalized (Λ(1) = 1). Thus Prob(X) is a closed convex subset of the unit ball of C(X)∗, it is compact and metrizable with respect to the weak-* topology defined by