Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane

2016/02/18 by Michael, Bialy, Andrey E. Mironov +1 · 2 citations
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #math.DG #math.DS #nlin.SI

paper · pdf · doi:10.48550/arxiv.1602.05698

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arxiv created 2016/02/18 · arxiv updated 2016/02/19

Abstract

We consider a convex curve γ 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 γ. We extend the result by S. Bolotin (1992) and get new obstructions on polynomial integrability in terms of the dual curve Γ. 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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