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Birkhoff-Poritsky conjecture for centrally-symmetric billiards

By Michael Bialy

Appears in 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.

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Citation data

  • DOI 10.24350/CIRM.V.19819703
  • Cite this video Bialy Michael (10/4/21). Birkhoff-Poritsky conjecture for centrally-symmetric billiards. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19819703
  • URL https://dx.doi.org/10.24350/CIRM.V.19819703

Bibliography

  • BIALY, Misha et MIRONOV, Andrey E. The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables. arXiv preprint arXiv:2008.03566, 2020. - https://arxiv.org/abs/2008.03566

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