---
title: "Analytic rigidity and symbolic dynamics for two-centre billiards"
authors:
  - "Stefano Baranzini"
  - "Susanna Terracini"
year: 2026
venue: "arXiv:math.DS"
arxiv: "2609.02310"
doi: null
source: "https://arxiv.org/abs/2609.02310"
topics:
  - "curiosity"
  - "maths-foundations"
seed_rank: 1620
seed_batch: "curiosity-2026-09-07"
reviewed: "2026-09-07"
pool: "maths-foundations"
relevance_score: 9
lineage: billiards
cites:
  - title: "Analytic rigidity and symbolic dynamics for two-centre billiards"
    url: "https://arxiv.org/abs/2609.02310"
    year: 2026
    arxiv: "2609.02310"
    doi: null
see:
  - "1583-symplectic-tiling-billiards-on-complete-affine-tori"
---

# Analytic rigidity and symbolic dynamics for two-centre billiards

## One-sentence takeaway

Sharp rigidity–chaos dichotomy for planar two-centre billiards: only confocal ellipses stay integrable at every energy — a Birkhoff–Poritsky analogue with two foci.

## Why it's lovely

Why you might love this: billiards again, but now two centres and a rigidity theorem with teeth. Continues the symplectic/tiling billiards delight from 1583 with a classical-mechanics punchline about when integrability survives.

## Key ideas

- Analogue of Birkhoff–Poritsky: integrable-at-every-energy tables should be ellipses confocal with the two centres.
- Analytic rigidity results for the integrable locus.
- Symbolic dynamics / chaos on the complementary side of the dichotomy.
- Planar two-centre billiards as the model system.

## Caveats

Analytic / smooth-category hypotheses on the table boundary; not a numerical recipe for chaotic billiard art.

## Implications for Broadside

Extends the billiards shelf (1583). Dynamical dichotomy is the kind of crisp mathematical story Broadside keeps for pure joy.

## Links

- arXiv abs: https://arxiv.org/abs/2609.02310
- PDF: https://arxiv.org/pdf/2609.02310.pdf