Counting problem on wind-tree models

TL;DR

This paper investigates the counting problem for closed billiard trajectories in periodic wind-tree models, providing asymptotic formulas and explicit constants that depend on obstacle geometry.

Contribution

It introduces asymptotic formulas for counting trajectories and computes the Siegel-Veech constant for generic wind-tree billiards, advancing understanding of their geometric and dynamical properties.

Findings

Asymptotic formulas for trajectory counts

Explicit Siegel-Veech constant depending on obstacle corners

Analysis of isotopy classes of billiard trajectories

Abstract

We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We show asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to $\mathbb{Z}^2$-translations) on the wind-tree billiard. We also compute explicitly the associated Siegel-Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.