Abstract
In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric -smooth convex planar billiards. We assume that the domain between the invariant curve of -periodic orbits and the boundary of the phase cylinder is foliated by -invariant curves. Under this assumption we prove that the billiard curve is an ellipse. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a -smooth foliation by convex caustics of rotation numbers in the interval , then the boundary curve is an ellipse. In the language of first integrals one can assert that if the billiard inside a centrally-symmetric -smooth convex curve admits a -smooth first integral with non-vanishing gradient on , then the curve is an ellipse.
The main ingredients of the proof are (1) the non-standard generating function for convex billiards; (2) the remarkable structure of the invariant curve consisting of -periodic orbits; and (3) the integral-geometry approach for rigidity results that was invented by the first named author for circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.
Citation
Misha Bialy. Andrey E. Mironov. "The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables." Ann. of Math. (2) 196 (1) 389 - 413, July 2022. https://doi.org/10.4007/annals.2022.196.1.2
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