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arXiv · 2504.03574

Non-Birkhoff periodic orbits in symmetric billiards

Casper Oelen·Bob Rink·Mattia Sensi
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Abstract

We study non-Birkhoff periodic orbits in symmetric convex planar billiards. Our main result provides a quantitative criterion for the existence of such orbits with prescribed minimal period, rotation number, and spatiotemporal symmetry. We exploit this criterion to find sufficient conditions for a symmetric billiard to possess infinitely many non-Birkhoff periodic orbits. It follows that arbitrarily small analytical perturbations of the circular billiard have non-Birkhoff periodic orbits of any rational rotation number and with arbitrarily long periods. We also generalize a known result for elliptical billiards to other $\mathbb{D}_2$-symmetric billiards. Lastly, we provide Matlab codes which can be used to numerically compute and visualize the non-Birkhoff periodic orbits whose existence we prove analytically.

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Casper Oelen, Bob Rink, Mattia Sensi. 2025-04-04. Non-Birkhoff periodic orbits in symmetric billiards. https://doi.org/10.1007/s00023-026-01670-7

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