Family 147Dynamical systems and ergodic theoryNo LeanNot yet digested
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.
Manuscripts in this family · 2
Continuous Phase Foliations Create Analytic Caustic Collars
Sep 24, 2026·paper.pdf
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.
Rigidity of Smooth Billiards with a Continuous Caustic Collar
Sep 24, 2026·paper.pdf
We prove that a smooth strictly convex planar billiard with positive curvature is an ellipse whenever a full neighborhood of its boundary is continuously foliated by smooth closed convex caustics. No differentiability across the leaves is assumed. This resolves the near-boundary Birkhoff conjecture in the stated smooth class.