Geometric Potentials and Pressure

This lecture focuses on introducing Banach spaces adapted to geometric potentials to prove a spectral gap for the transfer operator. It covers the existence and uniqueness of equilibrium states and the analyticity of the pressure function. The instructor discusses stable curves, the definition of pressure, norms, distances between curves and test functions, Lasota-Yorke unmatched pieces, regularity of the transfer operator, spectral decomposition, a spectral gap, entropy, uniqueness of equilibrium states, and a generalized variational principle. The lecture concludes with the analyticity of the pressure function and its derivatives, emphasizing the spectral gap and uniqueness of equilibrium measures.