A survey of results on ergodic Billiards
Roberto Markarian
We will present a survey of results and methods on the mathematical theory of billiards. We will focus on ergodic and statistical properties of classical billiard systems in the plane. The invariant measure for these systems was studied by Birkhoff in the twenties and the ergodic theory was developed by the soviet school in the sixties - seventies, specially by the seminal works of Jacob Sinai. Relations with the ergodic hypothesis of Boltzmann and properties of the geodesic flow were well understood from the very beginning. The survey will include references to my own work on ergodic and statistical properties and to recent results on decay of correlations and Poisson processes using methods by for example, Liverani, Demers, Baladi, Dolgopyat, Vaienti, Collet, Zhang, Melbourne.