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 (bt)t∈R on the plane with a Λ-periodic pattern (Λ⊂R2 is a lattice) of polygonal scatterers. We prove that if the billiard table is additionally rational then for a.e. direction θ∈S1 the Poisson suspension of the directional billiard flow (btθ)t∈R is weakly mixing. This gives the weak mixing of the Poisson suspension of (bt)t∈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 (btθ)t∈R is not mixing for a.e. θ∈S1.