Examples of minimal set for IFSs
A general law of large permanent
October 2017, 37(10): 5271-5284. doi: 10.3934/dcds.2017228
A locally integrable multi-dimensional billiard system
Dmitry Treschev *,
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
Figure(1)
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.
Keywords: Billiard system, integrable billiard system, multi-dimensional billiard system.
Mathematics Subject Classification: Primary: 37J10, 37J40; Secondary: 65P10.
Citation: Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
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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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Dmitry Treschev