A locally integrable multi-dimensional billiard system

Dmitry Treschev

doi:10.3934/dcds.2017228

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2017, Volume 37, Issue 10: 5271-5284. Doi: 10.3934/dcds.2017228

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A locally integrable multi-dimensional billiard system

Dmitry Treschev^*,

Steklov Mathematical Institute, 8 Gubkina St. Moscow, 119991, Russia

Received: December 31, 2016

Revised: April 30, 2017

Published: October 2017

The research is supported by the RNF grant 14-50-00005

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Abstract

We consider a multi-dimensional billiard system in an $(n+1)$-dimensional Euclidean space, the direct product of the "horizontal" hyperplane and the "vertical" line. The hypersurface that determines the system is assumed to be smooth and symmetric in all coordinate hyperplanes. Hence there exists a periodic orbit $γ$ of period 2 moving along the "vertical" coordinate axis. The question we ask is as follows. Is it possible to choose such a system to have the dynamics locally (near $γ$) conjugated to the dynamics of a linear map?

Since the problem is local, the billiard hypersurface can be determined as the graphs of the functions $± f$, where $f$ is even and defined in a neighborhood of the origin on the "horizontal" coordinate hyperplane. We prove that $f$ exists as a formal Taylor series in the non-resonant case and give numerical evidence for convergence of the series.

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Mathematics Subject Classification: Primary: 37J10, 37J40; Secondary: 65P10.

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Figure 1. The graph of $b^{-1/2}_\infty$ as a function of $\alpha/(2\pi)$. Two "gaps" correspond to the resonances $\frac\alpha{2\pi} = 3/10$ and $\frac\alpha{2\pi} = 1/3$

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