The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables


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. 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 with non-vanishing gradient 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; (2) the remarkable structure of the invariant curve consisting of $4$-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.


Misha Bialy

School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Israel

Andrey E. Mironov

Novosibirsk State University, Novosibirsk, Russia, and Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia