Thermodynamic Formalism for Dispersing Billiards

Explore the mathematical foundations of statistical mechanics through a lecture on thermodynamic formalism for dispersing billiards. Delve into recent findings on the existence and uniqueness of equilibrium states for geometric potentials in finite horizon dispersing billiards. Understand the significance of the potential family, with t=1 representing the smooth invariant (SRB) measure and t=0 corresponding to the measure of maximal entropy. Learn about the construction of anisotropic Banach spaces adapted to these potentials, leading to proofs of exponential mixing through spectral gaps in transfer operators. Examine the vanishing spectral gap as t approaches 0 and discuss the possibility of a phase transition at t=0. Enhance your understanding with accompanying slides that illustrate key concepts and mathematical formulations in this 48-minute presentation by Mark Demers for the International Mathematical Union.