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

TL;DR

This paper proves functional limit theorems for Birkhoff sums related to local times in the Z^d-periodic Lorentz gas with infinite horizon, revealing new stochastic properties of this infinite measure-preserving system.

Contribution

It establishes the first functional limit theorems for Birkhoff sums of local times in the infinite horizon Lorentz gas, extending understanding of ergodic properties in infinite measure systems.

Findings

Functional limit theorems for Birkhoff sums of local times.

Results apply to collision counts in different cells.

General convergence result for additive functionals of chaotic systems.

Abstract

This work is a contribution to the study of the ergodic and stochastic properties of Z^d-periodic dynamical systems preserving an infinite measure. We establish functional limit theorems for natural Birkhoff sums related to local times of the 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 Z^d-periodicity of the model we are interested in, these Birkhoff sums can be rewritten as additive functionals of a Birkhoff sum of the Sinai billiard. For completness and in view of future studies, we state a general result of convergence of additive functionals of Birkhoff sums of chaotic probability preserving dynamical systems under general assumptions.