On algebraically integrable Birkhoff and angular billiards

1 UMPA
UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
Abstract : We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We also extend this result to the case of piecewise-smooth and not necessarily convex polynomially integrable billiards: we show that the boundary is a union of confocal conical arcs and straight-line segments lying in some special lines defined by the foci. We also present a complexification of these results. The proof, which is obtained by Mikhail Bialy, Andrey Mironov and the author, is split into two parts. The first part is the paper by Bialy and Mironov, where they prove the following theorems: 1) the polar duality transforms a polynomially integrable planar billiard to a rationally integrable angular billiard; 2) the singularities and inflection points of each irreducible component of the complexified curve polar-dual to the billiard boundary lie in the two complex isotropic lines through the origin; 3) the Hessian Formula: appropriately defined Hessian of the integral of the angular billiard being restricted to the curve polar-dual to the boundary is a constant multiple of a power (x 2 + y 2) s. The present paper provides the second part of the proof. Namely, we prove that each irreducible component of the polar-dual curve that is not a line is a conic. This together with a theorem of S.V.Bolotin implies the main results: solution of the Algebraic Birkhoff Conjecture in both convex smooth and non-convex piecewise smooth cases.
Type de document :
Pré-publication, Document de travail
60 pages, 3 figures. A proof of Bolotin's theorem is added. A complexification of main results is.. 2017

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https://hal-ens-lyon.archives-ouvertes.fr/ensl-01664204
Contributeur : Alexey Glutsyuk <>
Soumis le : jeudi 14 décembre 2017 - 15:47:41
Dernière modification le : jeudi 11 janvier 2018 - 06:26:59

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int-birkhoff-hess.pdf
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• HAL Id : ensl-01664204, version 1
• ARXIV : 1706.04030

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Alexey Glutsyuk. On algebraically integrable Birkhoff and angular billiards. 60 pages, 3 figures. A proof of Bolotin's theorem is added. A complexification of main results is.. 2017. 〈ensl-01664204〉

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