# American Institute of Mathematical Sciences

October  2017, 37(10): 5271-5284. doi: 10.3934/dcds.2017228

## A locally integrable multi-dimensional billiard system

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

Received  January 2017 Revised  May 2017 Published  June 2017

Fund Project: The research is supported by the RNF grant 14-50-00005.

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.

Citation: Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
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##### References:
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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