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arXiv · 1602.05698

Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane

Michael·Bialy·Andrey E. Mironov
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Abstract

We consider a convex curve $\gamma$ lying on the Sphere or Hyperbolic plane. We study the problem of existence of polynomial in velocities integrals for Birkhoff billiard inside the domain bounded by $\gamma$. We extend the result by S. Bolotin (1992) and get new obstructions on polynomial integrability in terms of the dual curve $\Gamma$. We follow a method which was introduced by S. Tabachnikov for Outer billiards in the plane and was applied later on in our recent paper to Birkhoff billiards with the help of a new the so called Angular billiard.

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Michael, Bialy, Andrey E. Mironov. 2016-02-18. Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane. https://arxiv.org/abs/1602.05698

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