Integrable billiards on surfaces of constant curvature

Болотин, С. Н.

doi:10.1007/bf02102114

Cited by 23 publications

(44 citation statements)

References 4 publications

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“…The construction we propose is influenced from the classification theorems of Liouville surfaces given in [3,9,10,12,14] and the classical examples of integrable billiards described in §3 (see [1,2,4,5,[16][17][18]). The construction we propose is influenced from the classification theorems of Liouville surfaces given in [3,9,10,12,14] and the classical examples of integrable billiards described in §3 (see [1,2,4,5,[16][17][18]).…”

Section: Liouville Billiard Tablesmentioning

confidence: 99%

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Liouville billiard tables and an inverse spectral result

Popov

Topalov

2003

Ergod. Th. Dynam. Sys.

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We consider a class of billiard tables (X, g), where X is a smooth compact manifold of dimension two with smooth boundary ∂X and g is a smooth Riemannian metric on X, the billiard flow of which is completely integrable. The billiard table (X, g) is defined by means of a special double cover with two branched points and it admits a group of isometries G ∼ = Z 2 × Z 2 . Its boundary can be characterized by the string property; namely, the sum of distances from any point of ∂X to the branched points is constant. We provide examples of such billiard tables in the plane (elliptical regions), on the sphere S 2 , on the hyperbolic space H 2 , and on quadrics. The main result is that the spectrum of the corresponding Laplace-Beltrami operator with Robin boundary conditions involving a smooth function K on ∂X uniquely determines the function K, provided that K is invariant under the action of G.

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Section: Liouville Billiard Tablesmentioning

confidence: 99%

“…Integrable billiard tables on S 2 and H 2 have been considered earlier in [2,4]. Integrable billiard tables on S 2 and H 2 have been considered earlier in [2,4].…”

Section: 2mentioning

confidence: 99%

Liouville billiard tables and an inverse spectral result

Popov

Topalov

2003

Ergod. Th. Dynam. Sys.

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“…-Analytic integrability implies polynomial integrability, since each homogeneous part in P of an analytic integral is a first integral itself, see [32, p. 107] (the converse is obvious); -In the case, when Σ is a simply connected complete surface of constant curvature and the boundary ∂Ω is smooth and connected, polynomial integrability is equivalent to the existence of a polynomial integral as above in a neighborhood of the unit tangent bundle to ∂Ω in T Σ| Ω , by S.V.Bolotin's results [15,16,17], see Theorem 1.22 below. In this case each first integral that is just polynomial in P is globally analytic on T Σ, see [17, proof of proposition 2] and Theorem 1.22.…”

Section: Resultsmentioning

confidence: 99%

On Two-Dimensional Polynomially Integrable Billiards on Surfaces of Constant Curvature

Glutsyuk

2018

Dokl. Math.

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The polynomial version of Birkhoff Conjecture on integrable billiards on complete simply connected surfaces of constant curvature (plane, sphere, hyperbolic plane) was first stated, studied and solved in a particular case by Sergei Bolotin in 1990-1992. Here we present a complete solution of the polynomial version of Birkhoff Conjecture. Namely we show that every polynomially integrable real bounded planar billiard with C 2 -smooth connected boundary is an ellipse. We extend this result to billiards with piecewise-smooth and not necessarily convex boundary on arbitrary two-dimensional simply connected complete surface of constant curvature: plane, sphere, Lobachevsky-Poincaré (hyperbolic) plane; each of them being modeled as a plane or a (pseudo-) sphere in R 3 equipped with appropriate quadratic form. Namely, we show that a billiard is polynomially integrable, if and only if its boundary is a union of confocal conical arcs and appropriate geodesic segments. We also present a complexification of these results. These are joint results of Mikhail Bialy, Andrey Mironov and the author. The proof is split into two parts. The first part is given in two papers by Bialy and Mironov (in Euclidean and non-Euclidean cases respectively). Their geometric construction reduced the Polynomial Birkhoff Conjecture to a purely algebro-geometric problem to show that an irreducible algebraic curve in CP 2 with certain properties is a conic. They have shown that its singular and inflection points lie in the complex light conic of the above-mentioned quadratic form. In the present paper we solve the above algebro-geometric problem completely.

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“…Moreover, Inj(S 2 , g 0 ) = π. The dynamical billiards inside a convex domain of spheres with constant curvatures have been studied recently in [Bol92,Bia13,CP14]. Definition 1.…”

Section: Introductionmentioning

confidence: 99%

Convex billiards on convex spheres

Zhang¹

2017

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Abstract. In this paper we study the dynamical billiards on a convex 2D sphere. We investigate some generic properties of the convex billiards on a general convex sphere. We prove that C ∞ generically, every periodic point is either hyperbolic or elliptic with irrational rotation number. Moreover, every hyperbolic periodic point admits some transverse homoclinic intersections. A new ingredient in our approach is Herman's result on Diophantine invariant curves that we use to prove the nonlinear stability of elliptic periodic points for a dense subset of convex billiards.

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Integrable billiards on surfaces of constant curvature

Cited by 23 publications

References 4 publications

Liouville billiard tables and an inverse spectral result

Liouville billiard tables and an inverse spectral result

On Two-Dimensional Polynomially Integrable Billiards on Surfaces of Constant Curvature

Convex billiards on convex spheres

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