[#P3072] Birkhoff conjecture for integrable convex billiards
Problem. If a strictly convex \(C^\infty\) planar billiard table has an invariant essential caustic for every rotation number \(\rho\in(0,\tfrac12)\), must its boundary be an ellipse?
1Context
Known frontier: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Open boundary: The global classification of all smooth strictly convex integrable tables remains open.
2Problem setup
Definition 1 (billiard map). The boundary phase-space map taking one reflection to the next.
Definition 2 (integrable). The phase annulus is foliated by invariant essential curves, equivalently caustics in the stated formulation.
Remark 1. A caustic is a curve tangent to every segment of any orbit that starts tangent to it. Ellipses have a full foliation by confocal caustics; the conjecture says they are the only smooth strictly convex tables with this complete integrability.
3What counts as a solution
- Prove every table satisfying the invariant-caustic hypothesis is an ellipse.
- Or construct a nonelliptic smooth strictly convex table with the full caustic foliation.
1Status
Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.[1][2]
1Records
Notes and companion material
Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.
- Equivalent-formulation queries: Birkhoff billiard conjecture integrable ellipse open 2026; local strong Birkhoff conjecture almost every ellipse 2025
- Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses.
- Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.
How the 4 records connect
ProblemBirkhoff conjecture for integrable convex billiards
2See also
- Positive metric entropy for the standard mapdynamical systems
- Generic analytic Arnold diffusion in a priori stable systemsdynamical systems
- Persistent exponential stretching of a material line in two-dimensional Euler flowdynamical systems
How to cite
TheoremDB contributors, “Birkhoff conjecture for integrable convex billiards,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/birkhoff-billiard-conjectureThis page as plain text: birkhoff-billiard-conjecture.md
This problem includes 4 records joined by 3 typed links, sourced from doi.org[1], current as of August 1, 2026.
1References
- Packet source. Illya Koval, “Local strong Birkhoff conjecture and local spectral uniqueness of almost every ellipse”. Inventiones mathematicae 244(1) (2026), 221-298. DOI 10.1007/s00222-025-01397-y. main local strong Birkhoff theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves the local strong Birkhoff conjecture near almost every ellipse under the paper's perturbative hypotheses.Also cited at Illya Koval, “Local strong Birkhoff conjecture and local spectral uniqueness of almost every ellipse,” Inventiones Mathematicae 244(1) (2026), 221–298. main local strong Birkhoff theorem.Source used to assess the problem's recorded status.For Birkhoff conjecture for integrable convex billiards: This is the dated publication status for the canonical target Birkhoff conjecture for integrable convex billiards.Source named by the research packet.
- M. Bialy, C. Fierobe, A. Glutsyuk, M. Levi, A. Plakhov, and S. Tabachnikov, “Open problems on billiards and geometric optics”. arXiv:2110.10750 (2021). integrable billiards section. ↗preprint · reference source · arXiv:2110.10750, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Places Birkhoff's conjecture among current billiard problems and records known cases.Source used to assess the problem's recorded status.For Birkhoff conjecture for integrable convex billiards: Places Birkhoff's conjecture among current billiard problems and records known cases.
Original TheoremDB editorial statement and source synthesis; external works are used for citation only.