Counting problem on wind-tree models

Abstract

We study periodic wind-tree models, that is, billiards in the plane endowed with ${\mathbb{Z}}^2$–periodically located identical connected symmetric right-angled obstacles. We give 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 explicitly compute the associated Siegel–Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.