Generic properties of planar symplectic billiards

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.