# On Local Birkhoff Conjecture for Convex Billiards

@article{Kaloshin2016OnLB, title={On Local Birkhoff Conjecture for Convex Billiards}, author={Vadim Kaloshin and Alfonso Sorrentino}, journal={arXiv: Dynamical Systems}, year={2016} }

The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends the result in [3], where only unperturbed ellipses of small eccentricities were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards… Expand

#### 33 Citations

On polynomially integrable Birkhoff billiards on surfaces of constant curvature

- Mathematics
- 2017

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… Expand

Nearly Circular Domains Which Are Integrable Close to the Boundary Are Ellipses

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The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely… Expand

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

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The algebraic version of the Birkhoff conjecture is solved completely for billiards with a piecewise C2-smooth boundary on surfaces of constant curvature: Euclidean plane, sphere, and Lobachevsky… Expand

Non-smooth convex caustics for Birkhoff billiard

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This paper is devoted to the examination of the properties of the string construction for the Birkhoff billiard. Based on purely geometric considerations, string construction is suited to provide a… Expand

On commuting billiards in higher-dimensional spaces of constant curvature

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We consider two nested billiards in $\mathbb R^d$, $d\geq3$, with $C^2$-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines… Expand

Entropy of billiard maps and a dynamical version of the Birkhoff conjecture

- Mathematics
- 2018

Abstract We first prove that the simplest (smooth convex) billiard table is the circular one, in the sense that the associated billiard map has polynomial entropy equal to 1, while for all other… Expand

On the integrability of Birkhoff billiards

- Mathematics, Medicine
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2018

In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards,… Expand

A ug 2 01 8 On commuting billiards in higher dimensions

- 2018

We consider two nested billiards in Rn, n ≥ 3, with smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the… Expand

A survey on polynomial in momenta integrals for billiard problems

- Mathematics, Medicine
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2018

A short survey of recent results on the algebraic version of the Birkhoff conjecture for integrable billiards on surfaces of constant curvature and the existence of polynomial integrals for the two-sided magnetic billiard introduced by Kozlov and Polikarpov is given. Expand

On Local Integrability in Billiard Dynamics

- Mathematics, Computer Science
- Exp. Math.
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It is shown that the relative measure of the domain in the billiard phase space on which the dynamics is conjugated to the rigid rotation can reach 50%. Expand

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