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arXiv · 2606.10102

Existence of a Periodic Orbit for Billiards in Polygons

Giovanni Forni
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Abstract

We prove that the billiard flow in any finite polygon has at least one periodic orbit. The proof by contradiction is based on a fundamental result on the dynamics of the billiard flow by Galperin, Kr\"uger and Troubetzkoy, on the geometry of a one-parameter scaling of the natural Riemannian metric on the unit tangent bundle, and on the topology of the skeleton or cut-locus of the scaled metrics.

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Giovanni Forni. 2026-06-08. Existence of a Periodic Orbit for Billiards in Polygons. https://arxiv.org/abs/2606.10102

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