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arXiv · 2609.02310

Analytic rigidity and symbolic dynamics for two-centre billiards

Stefano Baranzini·Susanna Terracini
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Abstract

We establish a sharp rigidity--chaos dichotomy for planar two-centre billiards, motivated by a natural analogue of the Birkhoff--Poritsky conjecture: the only tables integrable at every energy should be ellipses confocal with the two centres. Let $\Omega$ be a bounded domain with $\mathcal C^1$ boundary containing the segment joining the centres. At every fixed energy $h\geq 0$, if $\partial\Omega$ is not a confocal ellipse, we construct billiard trajectories that shadow the stable and unstable manifolds of the collision--reflection orbit and realise arbitrarily prescribed sequences of sufficiently large winding numbers around the segment. This yields an invariant set semiconjugate to the full shift on a countable alphabet, periodic trajectories with prescribed finite itineraries, and compact invariant subsystems with arbitrarily large topological entropy. If, in addition, $\partial\Omega$ is real-analytic, every real-analytic function on the fixed-energy phase space $M_h$ that is invariant under the billiard map is constant. This establishes the real-analytic form of the two-centre Birkhoff--Poritsky conjecture throughout the non-negative-energy regime.

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Stefano Baranzini, Susanna Terracini. 2026-09-02. Analytic rigidity and symbolic dynamics for two-centre billiards. https://arxiv.org/abs/2609.02310

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