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arXiv · 2607.21223

Generic properties of planar symplectic billiards

Luca Baracco·Olga Bernardi·José Pedro Gaivão
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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.

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Luca Baracco, Olga Bernardi, José Pedro Gaivão. 2026-07-23. Generic properties of planar symplectic billiards. https://arxiv.org/abs/2607.21223

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