Soft Billiards with Corners

Abstract

We develop a framework for dealing with smooth approximations to billiards with corners in the two-dimensional setting. Let a polygonal trajectory in a billiard start and end up at the same billiard's corner point. We prove that smooth Hamiltonian flows which limit to this billiard have a nearby periodic orbit if and only if the polygon angles at the corner are “acceptable.” The criterion for a corner polygon to be acceptable depends on the smooth potential behavior at the corners, which is expressed in terms of a scattering function. We define such an asymptotic scattering function and prove the existence of it, explain how can it be calculated and predict some of its properties. In particular, we show that it is non-monotone for some potentials in some phase space regions. We prove that when the smooth system has a limiting periodic orbit it is hyperbolic provided the scattering function is not extremal there. We then prove that if the scattering function is extremal, the smooth system has elliptic periodic orbits limiting to the corner polygon, and, furthermore, that the return map near these periodic orbits is conjugate to a small perturbation of the Hénon map and therefore has elliptic islands. We find from the scaling that the island size is typically algebraic in the smoothing parameter and exponentially small in the number of reflections of the polygon orbit.