A locally integrable multi-dimensional billiard system

TL;DR

This paper investigates the local dynamics of a multi-dimensional billiard system near a symmetric periodic orbit, exploring conditions under which the system can be linearized and providing formal and numerical evidence for such linearization.

Contribution

It proves the existence of a formal Taylor series for the billiard hypersurface in the non-resonant case and offers numerical evidence for the convergence of this series.

Findings

Existence of a formal Taylor series for the hypersurface function f

Numerical evidence suggests the series converges in the non-resonant case

The system's local dynamics can be conjugated to a linear map near the periodic orbit

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 $\gamma$ 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 $\gamma$) 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 $\pm f$, 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.