The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables

Abstract

In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $\mathcal A$ between the invariant curve of $4$-periodic orbits and the boundary of the phase cylinder is foliated by $C^0$-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. Other versions of Birkhoff-Poritsky conjecture follow from this result. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a $C^1$-smooth foliation by convex caustics of rotation numbers in the interval (0; 1/4] then the boundary curve is an ellipse. In the language of first integrals one can assert that {if the billiard inside a centrally-symmetric $C^2$-smooth convex curve $\gamma$ admits a $C^1$-smooth first integral which is not singular on $\mathcal A$, then the curve $\gamma$ is an ellipse. } The main ingredients of the proof are : (1) the non-standard generating function for convex billiards discovered in [8], [10]; (2) the remarkable structure of the invariant curve consisting of $4$-periodic orbits; and (3) the integral-geometry approach initiated in [6], [7] for rigidity results of circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.