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Explore the fundamental concepts of entropy and Gibbs measures in hyperbolic systems through this comprehensive lecture. Delve into classical examples of hyperbolic systems, including toral automorphisms and geodesic flows. Examine entropy and pressure in terms of expansion in the unstable direction, and learn about the construction of SRB measures and Gibbs measures. Investigate the speed of mixing and resonances for hyperbolic systems, with a focus on explicit examples for toral automorphisms. Study classical results on entropy for geodesic flows and their partial analogues for translation surfaces. Conclude by exploring random walks in hyperbolic space, covering topics such as entropy, drift, and harmonic measures, and addressing the Kaimanovich-Le Prince question.
