Aperiodic and linearly repetitive Lorentz gases of finite horizon are not exponentially mixing

TL;DR

This paper demonstrates that certain aperiodic and linearly repetitive Lorentz gases with finite horizon do not exhibit exponential mixing, providing bounds on polynomial mixing speeds for specific observables.

Contribution

It establishes non-exponential mixing properties for a class of Lorentz gases and quantifies polynomial mixing speeds based on H"older regularity.

Findings

Lorentz gases are not exponentially mixing in any dimension.

Polynomial mixing speed bounds depend on H"older exponent.

Provides a comprehensive analysis of mixing rates for these systems.

Abstract

We prove that aperiodic and linearly repetitive Lorentz gases with finite horizon are not mixing with exponential or stretched exponential speed in any dimension for any class of H\"older observables. We also bound the polynomial speed of mixing for observables in the H\"older space $H_{\alpha}$ depending on $\alpha$.