Counting problem on infinite periodic billiards and translation surfaces

ABSTRACT

The Gauss circle problem consists in counting the number of integer points of bounded length in the plane. This problem is equivalent to counting the number of closed geodesics of bounded length on a flat two dimensional torus or, periodic trajectories, in a square billiard table.
Many counting problems in dynamical systems have been inspired by this problem. For 30 years, the experts try to understand the asymptotic behavior of closed geodesics in translation surfaces and periodic trajectories on rational billiards. (Polygonal billiards yield translation surfaces naturally through an unfolding procedure.) H. Masur proved that this number has quadratic growth rate.
In these talk, we will study the counting problem on infinite periodic rational billiards and translation surfaces.

Posted on Mar 28, 2019 in Dynamical Systems, Seminars