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Explore the intricacies of entropy and Gibbs measures for hyperbolic systems in this comprehensive lecture by Mark Pollicott from the University of Warwick. Delve into classical examples of hyperbolic systems, including toral automorphisms and geodesic flows, and understand entropy and pressure in terms of expansion in the unstable direction. Learn about the construction of SRB measures and Gibbs measures, drawing from the work of Sinai, Climenhaga, Pesin, and Zelerowicz. Investigate the speed of mixing and resonances for hyperbolic systems, with explicit examples for toral automorphisms. Examine classical results on entropy for geodesic flows, referencing the work of Katok, Manning, and Margulis, and discover a partial analogue for translation surfaces. Finally, explore random walks in hyperbolic space, focusing on entropy, drift, and harmonic measures, and consider the Kaimanovich-Le Prince question. This 49-minute lecture, part of the Simons Semester on Dynamics, offers a deep dive into advanced concepts in dynamical systems theory.
