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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {D}_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">D</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-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.</p>

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Non-Birkhoff periodic orbits in symmetric billiards

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月7日

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  • Casper Oelen,
  • Bob Rink,
  • Mattia Sensi

摘要

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 \(\mathbb {D}_2\) 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.