The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables

Abstract: In this talk (joint work with A.E. Mironov) I shall discuss a recent proof of the Birkhoff-Poritsky conjecture for centrally-symmetric C^2-smooth convex planar billiards. We assume that the domain 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. 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 for rigidity results of circular billiards. Surprisingly, our result yields a Hopf-type rigidity for billiard in ellipse.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

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