Mixing Property of Symmetrical Polygonal Billiards

TL;DR

This study numerically investigates the mixing properties of irrational polygonal billiards, providing evidence that certain symmetrical configurations exhibit strong mixing behavior characterized by power-law decay in autocorrelation functions.

Contribution

It introduces a biparametric family of polygonal billiards with rotational symmetries and analyzes their ergodic and mixing properties through numerical simulations.

Findings

Symmetrical polygonal billiards can be strongly mixing.

Power-law decay with exponent near 1 indicates strong mixing.

Higher symmetry parameter n tends toward integrable behavior.

Abstract

The present work consists of a numerical study of the dynamics of irrational polygonal billiards. Our contribution reinforces the hypothesis that these systems could be Strongly Mixing, although never demonstrably chaotic, and discuss the role of rotational symmetries on the billiards boundaries. We introduce a biparametric polygonal billiards family with only $C_n$ rotational symmetries. Initially, we calculate for some integers values of $n$ the filling of the phase space through the Relative Measure $r(\ell, \theta; t)$ for a plane of parameters $\ell \times \theta$. From the resulting phase diagram, we could identify the completely ergodic systems. The numerical evidence that symmetrical polygonal billiards can be Strongly Mixing is obtained by calculating the Position Autocorrelation Function, $\Cor_x(t)$, these figures of merit result in power law-type decays $t^{- \sigma}$. The Strongly Mixing property is indicated by $\sigma = 1$. For odd small values of $n$, the exponent $\sigma \simeq 1$ is obtained while $\sigma < 1$, weakly mixing cases, for small even values. Intermediate $n$ values present $\sigma \simeq 1$ independent of parity. For high values of symmetry parameter $n$, the biprametric family tends to be a circular billiard (integrable case). This range shows even less ergodic behavior when $n$ increases and $\sigma$ decreases.