Generic properties of planar symplectic billiards

Abstract

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 $C^{\infty }$ 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 $C^{\infty }$ strongly convex domains, elliptic periodic points are stable and hyperbolic periodic points have transverse homoclinic points. Based on these results, we conclude that --generically-- $C^{\infty }$ strongly convex symplectic billiard has positive topological entropy.

26 pages, comments are welcome