2026/07/23 by Luca Baracco, Olga Bernardi, José Pedro Gaivão
#math.DS
We study generic properties of planar symplectic billiards, a symplectic analogue of classical Birkhoff billiards introduced by P. Albers and S. Tabachnikov. We prove that, for a residual set of C^∞ strongly convex domains, periodic symplectic billiard trajectories are in general position and non-degenerate. We also prove a Franks' lemma for symplectic billiards and deduce that stably hyperbolic periodic points form a hyperbolic set given by the closure of the set of hyperbolic periodic points. We further show that, for a residual set of C^∞ strongly convex domains, elliptic periodic points are stable and hyperbolic periodic points have transverse homoclinic points. Based on these results, we conclude that --generically-- C^∞ strongly convex symplectic billiard has positive topological entropy.