Rates of mixing for the measure of maximal entropy of dispersing billiard maps

TL;DR

This paper studies the statistical properties of a special measure in dispersing billiard maps, showing it has polynomial to super-polynomial decay of correlations and satisfies the Central Limit Theorem, under certain recurrence conditions.

Contribution

It extends previous work by proving decay rates of correlations for the measure of maximal entropy in dispersing billiards, under a recurrence condition, with implications for generic billiard tables.

Findings

Polynomial decay of correlations for the measure of maximal entropy.

Super-polynomial decay for billiard tables with bounded complexity.

Validation of the Central Limit Theorem for observables under this measure.

Abstract

In a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for H\"older continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super-polynomial rate.