Equidistribution for standard pairs in planar dispersing billiard flows

TL;DR

This paper proves exponential decay of correlations in dispersing billiard flows on the 2-torus with finite horizon, focusing on initial measures on standard pairs and applications to heat conduction modeling.

Contribution

It extends correlation decay results to highly singular initial measures concentrated on standard pairs in dispersing billiards.

Findings

Exponential correlation decay for standard pairs in billiard flows.

Application to heat conduction models with singular initial measures.

Discussion of model dependence of decay bounds.

Abstract

We prove exponential correlation decay in dispersing billiard flows on the 2-torus assuming finite horizon and lack of corner points. With applications aimed at describing heat conduction, the highly singular initial measures are concentrated here on 1-dimensional submanifolds (given by standard pairs) and the observables are supposed to satisfy a generalized H\"older continuity property. The result is based on the exponential correlation decay bound of Baladi, Demers and Liverani obtained recentlyfor H\"older continuous observables in these billiards. The model dependence of the bounds is also discussed.