Decay of correlations for billiards with flat points II: cusps effect

TL;DR

This paper investigates the decay of correlations in a family of dispersing billiards with cusps, demonstrating polynomial decay rates and resolving an open question about mixing rates in such systems.

Contribution

It constructs a specific family of billiards with flat points and establishes their polynomial mixing rates, addressing an open problem in the field.

Findings

Correlation functions decay polynomially as O(n^{-1/(eta-1)})

The decay rate depends on the cusp parameter 2

Resolves an open question by Chernov and Markarian

Abstract

In this paper we continue to study billiards with flat points, by constructing a spec family of dispersing billiards with cusps. All boundaries of the table have positive curvature except that the curvature vanishes at the vertex of a cusp, i.e. there is a pair of boundaries intersection at the flat point tangentially. We study the mixing rates of this one-parameter family of billiards with parameter $\beta\in (2,\infty)$, and show that the correlation functions of the collision map decay polynomially with order $\cO(n^{-\frac{1}{\beta-1}})$ as $n\to\infty$. In particular, this solves an open question raised by Chernov and Markarian \cite{CM05}.