Limit theorems for Birkhoff sums and local times of the periodic Lorentz gas with infinite horizon

Abstract

This work is a contribution to the study of the ergodic and stochastic properties of dynamical systems preserving an infinite measure. We establish functional limit theorems for Birkhoff sums related to local times of the ${\mathbb{Z}}^d$-periodic Lorentz gas with infinite horizon, for both the collision map and the flow. In particular, our results apply to the difference between the numbers of collisions in two different cells. Because of the ${\mathbb{Z}}^d$-periodicity of the model we are interested in, these Birkhoff sums can be rewritten as additive functionals of Birkhoff sums of the Sinaǐ billiard. For completeness and in view of future studies, we state a general result of convergence for additive functionals of Birkhoff sums of chaotic probability preserving dynamical systems under general operator assumptions.