Chaotic properties for billiards in circular polygons

TL;DR

This paper investigates the complex chaotic behavior of billiard trajectories in circular polygon domains, revealing infinite entropy, many periodic orbits, and unique spectral properties through analytical methods.

Contribution

It establishes the existence of chaotic invariant sets, uncountably many asymptotic trajectories, and provides explicit bounds on periodic orbits in billiards within circular polygons.

Findings

Existence of a set with dynamics semiconjugate to a full shift, indicating chaos.

Uncountably many asymptotic trajectories approaching the boundary.

Exponential lower bound on the number of periodic trajectories as period increases.

Abstract

We study billiards in domains enclosed by circular polygons. These are closed $C^1$ strictly convex curves formed by finitely many circular arcs. We prove the existence of a set in phase space, corresponding to generic sliding trajectories close enough to the boundary of the domain, in which the return billiard dynamics is semiconjugate to a transitive subshift on infinitely many symbols that contains the full $N$-shift as a topological factor for any $N \in \mathbb{N}$, so it has infinite topological entropy. We prove the existence of uncountably many asymptotic generic sliding trajectories approaching the boundary with optimal uniform linear speed, give an explicit exponentially big (in $q$) lower bound on the number of $q$-periodic trajectories as $q \to \infty$, and present an unusual property of the length spectrum. Our proofs are entirely analytical.