Rafael Ramírez-Ros (Universitat Politècnica de Catalunya, Spain)
We find necessary and sufficient conditions for high-order persistence of
resonant caustics in perturbed circular billiards. The main technical tool
is a perturbation theory based on the Bialy-Mironov generating function
for convex billiards. A caustic is a curve such that any billiard
trajectory, once tangent to the curve, stays tangent after every reflection. A convex
caustic is p/q-resonant when all its tangent trajectories form closed
polygons with q sides that make p turns around the caustic. We prove
that any resonant caustic of the circular billiard with period q persist up to order
⌈q/n⌉ − 1 under any ‘polynomial’ perturbation of the circle of degree.
(This is a join work with Comlan Edmond Koudjinan.)