Bridge to Hyperbolic Polygonal Billiards

TL;DR

This paper demonstrates that physical polygonal billiards with a finite-sized particle are typically hyperbolic and have positive entropy, contrasting with mathematical billiards which are non-chaotic, and suggests they may also be ergodic.

Contribution

It shows that physical polygonal billiards are hyperbolic with positive entropy on a subset of positive measure, unlike mathematical billiards, and proposes their potential ergodicity.

Findings

Physical polygonal billiards are hyperbolic on a subset of positive measure.

They have positive Kolmogorov-Sinai entropy for any positive particle radius.

Typical physical billiards are equivalent to semi-dispersing billiards.

Abstract

It is well-known that billiards in polygons cannot be chaotic (hyperbolic). Particularly Kolmogorov-Sinai entropy of any polygonal billiard is zero. We consider physical polygonal billiards where a moving particle is a hard disc rather than a point (mathematical) particle and show that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov- Sinai entropy for any positive radius of the moving particle (provided that the particle is not so big that it cannot move within a polygon). This happens because a typical physical polygonal billiard is equivalent to a mathematical (point particle) semi-dispersing billiard. We also conjecture that in fact typical physical billiard in polygon is ergodic under the same conditions.