On some refraction billiards

The aim of this work is to continue the analysis, started in [10], of the dynamics of a point-mass particle P moving in a galaxy with an harmonic biaxial core, in whose center sits a Keplerian attractive center (e.g. a Black Hole). Accordingly, the plane R-2 is divided into two complementary domains, depending on whether the gravitational effects of the galaxy's mass distribution or of the Black Hole prevail. Thus, solutions alternate arcs of Keplerian hyperbol & AELIG; with harmonic ellipses; at the interface, the trajectory is refracted according to Snell's law. The model was introduced in [11], in view of applications to astrodynamics. In this paper we address the general issue of periodic and quasi-periodic orbits and associated caustics when the domain is a perturbation of the circle, taking advantage of KAM and Aubry-Mather theories.