Apparaît dans la collection : Differential Geometry, Billiards, and Geometric Optics / Géométrie différentielle, billards et optique géométrique
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.
De Barbara Schapira
De Barbara Schapira
