Based on joint work with Andrey E. Mironov (Novosibirsk).
In this talk I shall discuss Birkhoff-Poritsky conjecture for centrally symmetric $C^{2}$-smooth convex planar billiards. We assume that the domain. 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. 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 by the author for rigidity results of circular billiards.
Surprisingly, we establish Hopf-type rigidity for billiards in the ellipse.[-]