Asymptotic Homotopical Complexity of an Infinite Sequence of Dispersing $2D$ Billiards

TL;DR

This paper studies the complex chaotic behavior of an infinite sequence of dispersing 2D billiards, providing bounds and asymptotic formulas for rotation sets and entropies, revealing the system's large-scale topological structure.

Contribution

It introduces new bounds and asymptotic estimates for rotation sets and entropies in an infinite dispersing billiard system, advancing understanding of its chaotic dynamics.

Findings

Effective bounds for the rotation set R.

Compactness and density properties of the admissible rotation set AR.

Asymptotic formulas for topological and metric entropies.

Abstract

We investigate the large scale chaotic, topological structure of the trajectories of an infinite sequence of dispersing, hence ergodic, $2D$ billiards with the configuration space $Q_n=\mathbb{T}^2 \setminus \bigcup_{i=0}^{n-1} D_i$, where the scatterers $D_i$ ($i=0,1,\dots,n-1$) are disks of radius $r<<1$ centered at the points $(i/n, 0)$ mod $\mathbb{Z}^2$. We get effective lower and upper radial bounds for the rotation set $R$. Furthermore, we also prove the compactness of the admissible rotation set $AR$ and the fact that the rotation vectors $v$ corresponding to admissible periodic orbits form a dense subset of $AR$. We also obtain asymptotic lower and upper estimates for the sequence $h_{top}(n)$ of topological entropies and precise asymptotic formulas for the metric entropies $h_{\mu}(n,r)$.