Are all totally integrable outer-length billiards ellipses?
Let $K \subset \mathbb{R}^2$ be a compact convex body whose smooth boundary has positive curvature. Its outer length billiard map $T$ acts on $\mathbb{R}^2 \setminus K$ as follows. From a point $P$ draw the two support lines $\ell_1, \ell_2$ of $K$ through $P$, labelled so that travelling from $K$ towards $P$ along $\ell_1$ keeps $K$ on the left. Let $C$ be the circle outside $K$ that is tangent to both lines and touches $\ell_2$ at its point of contact with $K$, and let $\ell_3$ be the third common support line of $C$ and $K$. Then $T(P) = \ell_2 \cap \ell_3$.
Albers and Tabachnikov introduced this map; it is a positive twist map whose generating function is the sum of the lengths of the two tangent segments from $P$ to $\partial K$. Baracco, Bernardi and Fierobe showed that circles and ellipses are totally integrable: the exterior of an ellipse is foliated by invariant curves, the confocal ellipses. For the outer billiard, Bialy proved the corresponding rigidity: if the exterior of a $C^2$ convex curve of positive curvature is foliated by continuous non-contractible invariant curves, the curve is an ellipse. Bialy and Tabachnikov proved, for every $n \ge 3$, that there is a functional space of tables whose outer length billiard has an invariant curve consisting of $n$-periodic points.
Problem. If the entire exterior $\mathbb{R}^2 \setminus K$ is foliated by continuous $T$-invariant simple closed curves surrounding $K$, must $\partial K$ be an ellipse?
Source: Baracco, Bernardi and Fierobe (2025), §3.1, who ask whether circles and ellipses are the only totally integrable cases; Bialy and Mironov (2026), §9.3, Question (1). The outer length billiard was introduced by Albers and Tabachnikov (2024).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, integrable billiards, outer length billiards
References
@article{baracco2025,
author = {Baracco, Luca and Bernardi, Olga and Fierobe, Corentin},
title = {Starting the study of outer length billiards},
journal = {Pacific J. Math.},
volume = {339},
number = {2},
year = {2025},
pages = {201--222},
doi = {10.2140/pjm.2025.339.201},
eprint = {2504.11915},
archivePrefix = {arXiv},
}
@incollection{bialy2026icm,
author = {Bialy, Misha and Mironov, Andrey E.},
title = {Integrable billiards and related topics},
booktitle = {Proceedings of the International Congress of Mathematicians 2026, Volume 5: Invited Lectures (Sections 9--11)},
volume = {5},
year = {2026},
pages = {42--58},
doi = {10.1137/25M1804078},
eprint = {2510.03790},
archivePrefix = {arXiv},
}
@article{albers2024,
author = {Albers, Peter and Tabachnikov, Serge},
title = {Monotone twist maps and {D}owker-type theorems},
journal = {Pacific J. Math.},
volume = {330},
number = {1},
year = {2024},
pages = {1--24},
doi = {10.2140/pjm.2024.330.1},
eprint = {2307.01485},
archivePrefix = {arXiv},
}
@article{bialy2024outer,
author = {Bialy, Misha},
title = {Integrable outer billiards and rigidity},
journal = {J. Mod. Dyn.},
volume = {20},
year = {2024},
pages = {51--65},
doi = {10.3934/jmd.2024002},
eprint = {2306.12494},
archivePrefix = {arXiv},
}
@misc{bialy2026remarks,
author = {Bialy, Misha and Tabachnikov, Serge},
title = {Remarks on the outer length billiards},
year = {2026},
eprint = {2603.05998},
archivePrefix = {arXiv},
}
@misc{aim-outer-length-billiard-integrability,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Are all totally integrable outer-length billiards ellipses?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/265-outer-length-billiard-integrability.md},
note = {Entry 265 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00543, "Are all totally integrable outer-length billiards ellipses?", clopen.solutions.