Mixing property of symmetrical polygonal billiards.

Abstract

The present work consists of a numerical study of the dynamics of irrational polygonal billiards. Our contribution reinforces the hypothesis that these systems can be strongly mixing, although never demonstrably chaotic, and discusses the role of rotational symmetries on the billiards boundaries. We introduce a biparametric polygonal billiard family with only C_{n} rotational symmetries. Initially, we calculate through the relative measure r(ℓ,θ;t) the phase space filling. This is done for some integer values of n and for a plane of parameters ℓ×θ. From the resulting phase diagram, we can identify the fully ergodic systems. The numerical evidence that symmetrical polygonal billiards can be strongly mixing is obtained by evaluating the position autocorrelation function Cor_{x}(t), which follows a power-law-type decay t^{-σ}. The strongly mixing property is indicated by σ=1. For odd, small values of n, the exponent σ≃1 is found. On the other hand, σ<1 (weakly mixing cases) for small, even values of n. Intermediate n values present σ≃1 independently of parity. For larger values of symmetry parameter n, the biparametric family tends to be a circular billiard (integrable case). For such values of n, we identified even less ergodic behavior at the pace at which n increases and σ decreases.