Journal Article10.1134/S0081543816080022
On fourth-degree polynomial integrals of the Birkhoff billiard
M. Bialy,A. E. Mironov +1 more
- 01 Nov 2016
- Vol. 295, Iss: 1, pp 27-32
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TL;DR: In this article, the authors studied the Birkhoff billiard in a convex domain with a smooth boundary and showed that if this dynamical system has an integral which is polynomial in velocities of degree 4 and is independent with the velocity norm, then γ is an ellipse.
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Abstract: We study the Birkhoff billiard in a convex domain with a smooth boundary γ. We show that if this dynamical system has an integral which is polynomial in velocities of degree 4 and is independent with the velocity norm, then γ is an ellipse.
read more
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Citations
Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES
Anatole Katok,Boris Hasselblatt +1 more
- 01 Jan 1995
TL;DR: In this article, Katok and Mendoza introduced the concept of asymptotic invariants for low-dimensional dynamical systems and their application in local hyperbolic theory.
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On polynomially integrable Birkhoff billiards on surfaces of constant curvature
TL;DR: In this article, a solution of the algebraic version of Birkhoff Conjecture on integrable billiards is presented, where the boundary is a union of confocal conical arcs and appropriate geodesic segments.
21
On polynomially integrable planar outer billiards and curves with symmetry property
TL;DR: In this article, it was shown that every polynomially integrable planar outer convex billiard is elliptic, and an extension of this statement to non-convex billiards was shown.
18
Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane
M. Bialy,A. E. Mironov +1 more
TL;DR: In this article, it was shown that for any domain different from round disc for all but finitely many values of the magnitude of the magnetic field billiard motion does not have polynomial in velocities integral of motion.
15
References
Billiard map and rigid rotation
TL;DR: In this article, the authors proved that the answer to the question of whether a billiard map can be locally conjugated to a rigid rotation is positive in the category of formal series, and for good rotation angles the answer is also positive in an analytic category.
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Integrable curvilinear billiards
TL;DR: In this article, the existence of integrable two-dimensional billiards with curvilinear boundaries was investigated and a systematic method for the construction of such systems was developed.
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An integrable deformation of an ellipse of small eccentricity is an ellipse
TL;DR: In this paper, it was shown that the Birkhoff conjecture is true for small perturbations of ellipses of small eccentricity, which is a version of this conjecture.
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On conjugacy of convex billiards
TL;DR: In this paper, the authors show that if two billiard maps are $C^{1,\alpha}$-conjugate near the boundary, for some α > 1/2, then the corresponding domains are similar, i.e., they can be obtained one from the other by rescaling and an isometry.
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Angular Billiard and Algebraic Birkhoff conjecture
Michael Bialy,Andrey E. Mironov +1 more
TL;DR: In this article, a new dynamical system called Angular billiard is introduced, which acts on the exterior points of a convex curve in Euclidean plane and is dual to the Birkhoff billiard.