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BIRS Workshop Lecture Videos
Asymptotic formulas on infinite periodic translation surfaces. Pardo, Angel
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. 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 this talk, we will study the counting problem on infinite periodic rational billiards and translation surfaces. The first example and motivation is the wind-tree model, a Z^2-periodic billiard model. In the classical setting, we place identical rectangular obstacles in the plane at each integer point; we play billiard on the complement. It is possible to give quite precise results on the counting problem for this model, thanks to the many symmetries it presents. These results, however, do not extend to more general contexts. I will present a general result on the counting problem for infinite periodic translation surfaces that uses new ideas: a dynamical analogous, for the algebraic hull of a cocycle, to strong and super-strong approximation on algebraic groups. Under these approximation hypothesis I will exhibit asymptotic formulas for the number of closed geodesics of bounded length on infinite periodic translation surfaces. And will present some applications and discuss why I think these hypothesis hold in general (work in progress).
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