A functional analytic approach to perturbations of the Lorentz gas

TL;DR

This paper develops a functional analytic framework using transfer operator spectra to analyze various perturbations of the Lorentz gas billiard system, ensuring spectral stability and enabling limit theorems.

Contribution

It introduces a flexible Banach space framework that accommodates diverse perturbations of the Lorentz gas, preserving spectral gaps and facilitating analysis.

Findings

Spectral gaps are stable under broad perturbations.

The framework applies to external forces and nonelastic reflections.

Limit theorems remain valid for perturbed systems.

Abstract

We present a functional analytic framework based on the spectrum of the transfer operator to study billiard maps associated with perturbations of the periodic Lorentz gas. We show that recently constructed Banach spaces for the billiard map of the classical Lorentz gas are flexible enough to admit a wide variety of perturbations, including: movements and deformations of scatterers; billiards subject to external forces; nonelastic reflections with kicks and slips at the boundaries of the scatterers; and random perturbations comprised of these and possibly other classes of maps. The spectra and spectral projections of the transfer operators are shown to vary continuously with such perturbations so that the spectral gap enjoyed by the classical billiard persists and important limit theorems follow.