Sub-Riemannian geometry and periodic orbits in classical billiards

Abstract

Classical (Birkhoff) billiards with full 1-parameter families of periodic orbits are considered. It is shown that construction of a convex billiard with a caustic (i.e. carrying only periodic orbits ) can be reformulated as the problem of finding a closed curve tangent to a non-integrable distribution on a manifold. The properties of this distribution are described as well as the consequences for the billiards with rational caustics. A particular implication of this construction is that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist.