Track billiards

TL;DR

This paper investigates a class of planar and 3D billiards with invariant phase space structures, demonstrating conditions under which they exhibit hyperbolic dynamics and exploring the mechanisms behind their chaotic behavior.

Contribution

It introduces a new class of track billiards with specific geometric properties and proves their hyperbolicity, extending results to three dimensions and analyzing the underlying chaotic mechanisms.

Findings

Billiards have non-zero Lyapunov exponents almost everywhere.

Invariant phase space sets are characterized for these billiards.

Hyperbolicity can arise without the defocusing mechanism in some subclasses.

Abstract

We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the segments and the arcs, the billiards considered have non-zero Lyapunov exponents almost everywhere. These results are then extended to a similar class of of 3-dimensional billiards. Finally, we find that for some subclasses of track billiards, the mechanism generating hyperbolicity is not the defocusing one that requires every infinitesimal beam of parallel rays to defocus after every reflection off of the focusing boundary.