Browse › Dynamical systems and ergodic theory › 147 The near-boundary Birkhoff conjecture
Continuous Phase Foliations Create Analytic Caustic Collars
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Abstract
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.
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Possibly relevant literature (found by the agent, not checked)
- Marie-Claude Arnaud and Maxime Zavidovique, Actions of symplectic homeomorphisms/diffeomorphisms on foliations by curves in dimension 2 (arXiv:2011.01546, 2020-11-02). Its results on generating-function regularity and transverse Hölder regularity for continuous invariant foliations of symplectic twist maps provide background for the claimed passage from continuous phase leaves to a jointly analytic caustic collar.
- Guan Huang, Vadim Kaloshin, Alfonso Sorrentino, Nearly circular domains which are integrable close to the boundary are ellipses (arXiv:1705.10601, 2017; to appear in GAFA per arXiv comments). A perturbative result under integrability near the boundary for domains close to low-eccentricity ellipses; relevant to the paper's statement that its reduction needs no closeness to a circle or ellipse, and not in its bibliography.
- Misha Bialy, Andrey E. Mironov, Integrable Billiards and Related Topics (arXiv:2510.03790, October 2025). A survey whose open questions (Section 9.1) ask whether their centrally symmetric rigidity theorem holds with a smaller neighbourhood of the boundary; relevant to the paper's comparison with Bialy–Mironov (2022).
- Klaudiusz Czudek, Jacopo De Simoi, Andrew Gad, Marco Poon, Yet another characterization of Birkhoff Billiards inside discs (arXiv:2603.16468, March 2026). Characterizes circular tables through the expansion of the formal Lazutkin conjugacy at the boundary; possibly relevant to the paper's use of grazing (near-boundary) normalization.
Digests and expositions
- heyprof, Birkhoff billiards: only the ellipse has no gaps (interactive exposition, undated). Illustrates caustics and phase portraits for ellipses and deformed convex tables, explicitly relating a full near-boundary foliation to the claims of family 147. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 147
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026,
author = {{OpenAI}},
title = {{Continuous Phase Foliations Create Analytic Caustic Collars}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026/paper.pdf}{OAI:Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026}},
year = {2026}
}Family 147: The near-boundary Birkhoff conjecture
Resolves the near-boundary Birkhoff conjecture for smooth strictly convex planar billiards of positive curvature. Such a billiard is an ellipse whenever a full grazing annulus is continuously foliated by individually invariant essential curves. A continuous physical collar of smooth closed convex caustics also suffices.
Other papers in this family:
- Rigidity of Smooth Billiards with a Continuous Caustic Collar · September 24, 2026