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SUMMARY:Misha Bialy (Tel Aviv University)
DTSTART:20210211T150000Z
DTEND:20210211T160000Z
DTSTAMP:20260423T023016Z
UID:DinAmicI/17
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/DinAmicI/17/
 ">Birkhoff-Poritsky conjecture for centrally-symmetric billiards</a>\nby M
 isha Bialy (Tel Aviv University) as part of DinAmicI: Another Internet Sem
 inar\n\n\nAbstract\nIn this talk I shall discuss Birkhoff-Poritsky conject
 ure for centrally-symmetric $C^2$-smooth convex planar billiards. \n\nWe a
 ssume that the domain $\\mathcal A$ between the invariant curve of $4$-per
 iodic orbits and the boundary of the phase cylinder is foliated by $C^0$-i
 nvariant curves.  \nUnder this assumption we prove that the billiard curve
  is an ellipse.\nOther versions of Birkhoff-Poritsky conjecture follow fro
 m this result. \nFor the original Birkhoff-Poritsky formulation we show th
 at if a neighborhood of the boundary of billiard domain has\na $C^1$-smoot
 h foliation by convex caustics of rotation numbers in the interval (0\; 1/
 4]\nthen the boundary curve is an ellipse. \n\nThe main ingredients of the
  proof are:\n\n \n(1) the non-standard generating function for convex bill
 iards\;  \n\n(2) the remarkable structure of the invariant curve consistin
 g of $4$-periodic orbits\; and\n\n \n(3) the integral-geometry approach in
 itiated in\n\\cite{B0}\, \\cite{B1} for rigidity results of circular billi
 ards. \n\n\nSurprisingly\, we establish a Hopf-type rigidity for billiards
  in the ellipse.\nBased on a joint work with Andrey E. Mironov (Novosibirs
 k).\n
LOCATION:https://researchseminars.org/talk/DinAmicI/17/
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