Banach Spaces and Thermodynamic Formalism

This lecture focuses on introducing Banach spaces adapted to the potential for the measure of maximal entropy in dispersing billiard maps. The instructor discusses modifications to prove the existence and uniqueness of the measure of maximal entropy, despite the inability to prove spectral gap for the transfer operator. Various concepts such as the transfer operator, stable curves, topological entropy, and the construction of the measure of maximal entropy are covered. The lecture also delves into the definition of norms, including weak and strong norms, and their implications in the context of billiard maps. The instructor explores the bounds on the spectral radius of the transfer operator and the construction of eigen measures. The lecture concludes with discussions on the hyperbolicity, ergodicity, mixing properties, entropy, uniqueness of measures, and open questions in the field.