# On polynomially integrable planar outer billiards and curves with symmetry property

@article{Glutsyuk2016OnPI, title={On polynomially integrable planar outer billiards and curves with symmetry property}, author={A. A. Glutsyuk and Eugenii Shustin}, journal={Mathematische Annalen}, year={2016}, volume={372}, pages={1481-1501} }

We show that every polynomially integrable planar outer convex billiard is elliptic. We also prove an extension of this statement to non-convex billiards.

## 13 Citations

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

- MathematicsDoklady Mathematics
- 2018

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…

### On rationally integrable planar dual multibilliards and piecewise smooth projective billiards

- Mathematics
- 2023

A planar projective billiard is a planar curve C equipped with a transversal line field. It defines reflection of lines from C. Its projective dual is a dual billiard: a curve γ ⊂ RP equipped with a…

### On polynomially integrable Birkhoff billiards on surfaces of constant curvature

- MathematicsJournal of the European Mathematical Society
- 2020

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…

### Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane

- MathematicsRussian Mathematical Surveys
- 2019

Magnetic billiards in a convex domain with smooth boundary on a constant-curvature surface in a constant magnetic field is considered in this paper. The question of the existence of an integral of…

### On infinitely many foliations by caustics in strictly convex open billiards

- Mathematics
- 2021

Reﬂection in strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve C whose tangent lines are reﬂected by the billiard to…

### Outer Billiards with the Dynamics of a Standard Shift on a Finite Number of Invariant Curves

- MathematicsExp. Math.
- 2021

Abstract We give a beautiful explicit example of a convex plane curve such that the outer billiard has a given finite number of invariant curves. Moreover, the dynamics on these curves is a standard…

### The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables

- Mathematics
- 2020

In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $\mathcal A$ between the invariant curve of…

### On Local Integrability in Billiard Dynamics

- MathematicsExp. Math.
- 2019

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%.

### A locally integrable multi-dimensional billiard system

- Mathematics
- 2016

We consider a multi-dimensional billiard system in an \begin{document}$(n+1)$\end{document} -dimensional Euclidean space, the direct product of the "horizontal" hyperplane and the "vertical" line.…

### Open problems, questions and challenges in finite- dimensional integrable systems

- MathematicsPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2018

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of…

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### 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…

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### On polynomially integrable Birkhoff billiards on surfaces of constant curvature

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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…