{
  "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 $\\mathcal C^1$ boundary containing the segment joining the centres. At every fixed energy $h\\geq 0$, if $\\partialΩ$ 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, $\\partialΩ$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02310",
  "author_total": 2,
  "authors": [
    {
      "name": "Stefano Baranzini"
    },
    {
      "name": "Susanna Terracini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6y94zxm24e4w6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02310",
  "fulltext_url": null,
  "id": "6y94zxm24e4w6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02310",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7207757994",
  "pdf_url": "https://arxiv.org/pdf/2609.02310",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6y94zxm24e4w6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02310",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02310"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Analytic rigidity and symbolic dynamics for two-centre billiards",
  "title_text": "Analytic rigidity and symbolic dynamics for two-centre billiards",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/6y94zxm24e4w6",
  "venue": "arXiv",
  "versions": []
}