Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows

TL;DR

This paper proves that dispersing billiard flows with cusps exhibit rapid mixing with correlations decaying faster than any polynomial rate, leading to strong statistical properties unlike the billiard map.

Contribution

It establishes rapid mixing and invariance principles for dispersing billiard flows with cusps, improving understanding of their statistical behavior.

Findings

Flow correlations decay faster than any polynomial rate

Flow admits the almost sure invariance principle

Results extend to Bunimovich flowers and stadia

Abstract

Following recent work of Chernov, Markarian, and Zhang, it is known that the billiard map for dispersing billiards with zero angle cusps has slow decay of correlations with rate 1/n. Since the collisions inside a cusp occur in quick succession, it is reasonable to expect a much faster decay rate in continuous time. In this paper we prove that the flow is rapid mixing: correlations decay faster than any polynomial rate. A consequence is that the flow admits strong statistical properties such as the almost sure invariance principle, even though the billiard map does not. The techniques in this paper yield new results for other standard examples in planar billiards, including Bunimovich flowers and stadia.