Definition and basic property of measure theoretic dynamical entropy (elementary and conditionally). Ergodic theorem for entropy. Topological entropy and variational principle. Measures of maximal entropy. Equidistribution of periodic points. Measure rigidity for commuting maps on the circle group.
The course will lead to a firm understanding of measure theoretic dynamical entropy and its applications within dynamics. We will start with the basic properties of (conditional) entropy, relate it to the question of effective coding techniques, discuss and prove the Shannon-McMillan-Breiman theorem that is also known as the ergodic theorem for entropy. Moreover, we will discuss a topological counter part and relate this topological entropy to the measure theoretic entropy by the variational principle. We will use these methods to classify certain natural homogeneous measures, prove equidistribution of periodic points on compact quotients of hyperbolic surfaces, and establish a measure rigidity theorem for commuting maps on the circle group.