Analytic rigidity and symbolic dynamics for two-centre billiards
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 Ω be a bounded domain with 𝒞1 boundary containing the segment joining the centres. At every fixed energy h≥ 0, if ∂Ω 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, ∂Ω is real-analytic, every real-analytic function on the fixed-energy phase space Mh 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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BibTeX
@misc{baranzini2026analytic,
title = {{Analytic rigidity and symbolic dynamics for two-centre billiards}},
author = {Stefano Baranzini and Susanna Terracini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02310},
eprint = {2609.02310},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02310},
}
RIS
TY - UNPB
TI - Analytic rigidity and symbolic dynamics for two-centre billiards
AU - Stefano Baranzini
AU - Susanna Terracini
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02310
UR - https://arxiv.org/abs/2609.02310
AB - 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 Ω be a bounded domain with 𝒞^1 boundary containing the segment joining the centres. At every fixed energy h≥ 0, if ∂Ω 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, ∂Ω 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.
ER -