Quantitative Mixing for Anosov Flows and the Decay of Correlations in Hyperbolic Billiards

This monograph presents a rigorous analysis of the quantitative mixing properties of Anosov flows, with a specific focus on the decay of correlations in hyperbolic billiards, such as the Sinai billiard. We bridge the gap between the smooth theory of contact Anosov flows, where Dolgopyat's estimates provide exponential decay rates, and the singular dynamics inherent to billiard systems. By employing a functional analytic framework based on transfer operators acting on anisotropic Banach spaces, we derive spectral gap properties that necessitate exponential mixing for finite horizon configurations. The study further elucidates the role of non-integrability of stable and unstable foliations in establishing these quantitative bounds. We review the transition from symbolic dynamics and Markov partitions to direct operator-theoretic methods, highlighting recent advances in handling the singularities arising from grazing collisions.