Birkhoff-Poritsky conjecture for centrally-symmetric billiards

Abstract: In this talk I shall discuss 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.

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 initiated in \cite{B0}, \cite{B1} for rigidity results of circular billiards.

Surprisingly, we establish a Hopf-type rigidity for billiards in the ellipse. Based on a joint work with Andrey E. Mironov (Novosibirsk).

dynamical systems

Audience: researchers in the topic

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