Basic Ergodic Theory

Definition 1. (Measure-preserving dynamical system) A measure-preserving dynamical system \((X, \mathcal{X}, \mu, T)\) is defined by specifying a probability space \((X, \mathcal{X}, \mu)\) and a measure-preserving transformation (isomorphism) \(T\) on it:

• \(X\) is a set.
• \(\mathcal{X}\) is a σ-algebra over \(X\).
• \(\mu : \mathcal{X} \to [0,1]\) is a probability measure.
• \(T : X \to X\) is a measurable transformation which preserves the measure \(\mu\), i.e., \(\forall E \in \mathcal{X}, \mu(T(E)) = \mu(E)\).

Definition 2. (Measure-theoretic entropy; Kolmogorov-Sinai entropy) The entropy of a partition \(Q=\{Q_1,\dots,Q_k\}\) is defined as \[\operatorname{H}(Q) = -\sum_{m=1}^k \mu(Q_m) \log \mu(Q_m)\] The measure-theoretic entropy of a dynamical system \((X,\mathcal{X},T,\mu)\) with respect to a partition \(Q=\{Q_1,\dots,Q_k\}\) is defined as \[h_\mu(T,Q) = \lim_{N \to \infty} \frac{1}{N} \operatorname{H}\left(\bigvee_{n=0}^N T^{-n}Q\right)\] The measure-theoretic entropy of a dynamical system \((X,\mathcal{X},T,\mu)\) is defined as \[h_\mu(T) = \sup_Q h_\mu(T,Q)\] where the supremum is taken over all finite measurable partitions \(Q\).