OP-00666 Solution claimed

Global Birkhoff billiard integrability conjecture

Let $D \subset \mathbb{R}^2$ be a bounded strictly convex domain whose boundary is a single $C^\infty$ curve with everywhere positive curvature. Its billiard map $T$ sends a boundary point with an inward direction to the next point where the ray meets $\partial D$, with the direction reflected specularly. Its phase space is the annulus of pairs $(s, \varphi)$, where $s$ is arclength along $\partial D$ and $\varphi \in (0, \pi)$ is the angle from the positive tangent; $\varphi = 0$ is the grazing boundary. In an ellipse the confocal ellipses are caustics. Bialy proved that if the whole phase annulus is foliated by continuous invariant curves that are not null-homotopic, then $D$ is a disk. Bialy and Mironov proved that a centrally symmetric $C^2$ convex table is an ellipse if the region between the invariant curve of $4$-periodic orbits and the boundary of the phase annulus is foliated by continuous invariant curves; Kaloshin, Koudjinan and Zhang proved a perturbative version for domains close to centrally symmetric ones, and Koval proved that a small perturbation of almost every ellipse that preserves integrability near the boundary is itself an ellipse.

Problem. Suppose an open annular neighbourhood of the grazing boundary $\varphi = 0$ is foliated by smooth $T$-invariant essential graphs $\varphi = \varphi_c(s)$, so that a full neighbourhood, including its rational rotation levels, is integrable. Must $\partial D$ be an ellipse? No assumption that $D$ is close to an ellipse or centrally symmetric is allowed.

Source: Birkhoff (1927) and Poritsky (1950), for integrable billiards; stated in this near-boundary form by AIM — Open Applied Problems, after Kaloshin, Koudjinan and Zhang (2024) and Koval (2026).

Imported from github.com/MColbrook/AIM.

Classification

Primary: DS — Dynamical Systems
Secondary: MG
Keywords: AIM Open Applied Problems, Birkhoff conjecture, caustics, convex billiards

References

@article{birkhoff1927,
  author  = {Birkhoff, George D.},
  title   = {On the periodic motions of dynamical systems},
  journal = {Acta Math.},
  volume  = {50},
  year    = {1927},
  pages   = {359--379},
  doi     = {10.1007/BF02421325},
}
@article{poritsky1950,
  author  = {Poritsky, Hillel},
  title   = {The billard ball problem on a table with a convex boundary -- an illustrative dynamical problem},
  journal = {Ann. of Math. (2)},
  volume  = {51},
  number  = {2},
  year    = {1950},
  pages   = {446--470},
  doi     = {10.2307/1969334},
}
@article{bialy1993,
  author  = {Bialy, Misha},
  title   = {Convex billiards and a theorem by {E}. {H}opf},
  journal = {Math. Z.},
  volume  = {214},
  number  = {1},
  year    = {1993},
  pages   = {147--154},
  doi     = {10.1007/BF02572397},
}
@article{bialy2022,
  author        = {Bialy, Misha and Mironov, Andrey E.},
  title         = {The {B}irkhoff--{P}oritsky conjecture for centrally-symmetric billiard tables},
  journal       = {Ann. of Math. (2)},
  volume        = {196},
  number        = {1},
  year          = {2022},
  pages         = {389--413},
  doi           = {10.4007/annals.2022.196.1.2},
  eprint        = {2008.03566},
  archivePrefix = {arXiv},
}
@article{kaloshin2024,
  author        = {Kaloshin, Vadim and Koudjinan, Comlan Edmond and Zhang, Ke},
  title         = {Birkhoff conjecture for nearly centrally symmetric domains},
  journal       = {Geom. Funct. Anal.},
  volume        = {34},
  number        = {6},
  year          = {2024},
  pages         = {1973--2007},
  doi           = {10.1007/s00039-024-00695-6},
  eprint        = {2306.12301},
  archivePrefix = {arXiv},
}
@article{koval2026,
  author        = {Koval, Illya},
  title         = {Local strong {B}irkhoff conjecture and local spectral uniqueness of almost every ellipse},
  journal       = {Invent. Math.},
  volume        = {244},
  number        = {1},
  year          = {2026},
  pages         = {221--298},
  doi           = {10.1007/s00222-025-01397-y},
  eprint        = {2111.12171},
  archivePrefix = {arXiv},
}
@misc{aim-birkhoff-billiard-integrability,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Global Birkhoff billiard integrability conjecture}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/064-birkhoff-billiard-integrability.md},
  note         = {Entry 064 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Continuous Phase Foliations Create Analytic Caustic Collars v1 GitHub ↗

OpenAI

For a smooth strictly convex planar billiard of positive curvature, a continuous foliation of a full grazing annulus by individually invariant essential curves forces an analytic boundary and a jointly analytic collar of smooth strictly convex caustics. The physical-collar rigidity theorem proved in the companion article then implies that the table is an ellipse.

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/main.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00666, "Global Birkhoff billiard integrability conjecture", clopen.solutions.