Ergodic properties of the ideal gas model for infinite billiards

Abstract

In this paper we study ergodic properties of the Poisson suspension (the ideal gas model) of the billiard flow $(b_t)_{t\in\mathbb R}$ on the plane with a $\Lambda$-periodic pattern ($\Lambda\subset\mathbb R^2$ is a lattice) of polygonal scatterers. We prove that if the billiard table is additionally rational then for a.e. direction $\theta\in S^1$ the Poisson suspension of the directional billiard flow $(b^\theta_t)_{t\in\mathbb R}$ is weakly mixing. This gives the weak mixing of the Poisson suspension of $(b_t)_{t\in\mathbb R}$. We also show that for a certain class of such rational billiards (including the periodic version of the classical wind-tree model) the Poisson suspension of $(b^\theta_t)_{t\in\mathbb R}$ is not mixing for a.e. $\theta\in S^1$.