On convex planar billiards, Birkhoff Conjecture and whispering galleries

A mathematical billiard is a system describing the inertial motion of a point mass inside a domain with elastic reflections at the boundary. In the case of convex planar domains, this model was first introduced and studied by G.D. Birkhoff, as a paradigmatic example of a low dimensional conservative dynamical system. A very interesting aspect is represented by the presence of 'caustics', namely curves inside the domain with the property that a trajectory, once tangent to it, stays tangent after every reflection (as on the right Figure). Besides their mathematical interest, these objects can explain a fascinating acoustic phenomenon, known as "whispering galleries", which can be sometimes noticed beneath a dome or a vault. The classical Birkhoff conjecture states that the only integrable billiard, i.e., the one having a region filled with caustics, is the billiard inside an ellipse. We show that this conjecture holds near ellipses.