TY - JOUR

T1 - The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables

AU - Bialy, Misha

AU - Mironov, Andrey E.

N1 - Funding Information: Keywords: Birkhoff-Poritsky conjecture, integrable billiard, Birkhoff billiard AMS Classification: Primary: 37J35, 37J40, 37J51. MB was partially supported by ISF grant 580/20. AEM was supported by Mathematical Center in Akademgorodok under agreement No 075-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation. © 2022 Department of Mathematics, Princeton University. Publisher Copyright: © 2022. Department of Mathematics, Princeton University.

PY - 2022/7

Y1 - 2022/7

N2 - In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric C2-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 C0-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 C1-smooth foliation by convex caustics of rotation numbers in the interval (0, 1/4], then the boundary curve is an ellipse. In the language of first integrals one can assert that if the billiard inside a centrally-symmetric C2-smooth convex curve admits a C1-smooth first integral with non-vanishing gradient on A, then the 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 for rigidity results that was invented by the first named author for circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.

AB - In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric C2-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 C0-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 C1-smooth foliation by convex caustics of rotation numbers in the interval (0, 1/4], then the boundary curve is an ellipse. In the language of first integrals one can assert that if the billiard inside a centrally-symmetric C2-smooth convex curve admits a C1-smooth first integral with non-vanishing gradient on A, then the 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 for rigidity results that was invented by the first named author for circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.

KW - Birkhoff billiard

KW - Birkhoff-poritsky conjecture

KW - Integrable billiard

UR - http://www.scopus.com/inward/record.url?scp=85131931148&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/74b8d3ec-9f28-3ab6-83c0-b28a5c99d83c/

U2 - 10.4007/ANNALS.2022.196.1.2

DO - 10.4007/ANNALS.2022.196.1.2

M3 - Article

AN - SCOPUS:85131931148

VL - 196

SP - 389

EP - 413

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 1

ER -