A family of chaotic billiards with variable mixing rates

TL;DR

This paper introduces a family of dispersing billiards with a tunable parameter that controls the decay rate of correlations, providing explicit relations between the parameter and the mixing properties.

Contribution

It presents a novel family of billiards where the mixing rate varies continuously with a parameter, and derives explicit formulas linking the parameter to the decay degree.

Findings

Correlation decay rate varies as 1/n^a with a in (1, ∞)

Explicit relation between the parameter and decay degree a

Family exhibits hyperbolic, ergodic, and mixing properties

Abstract

We describe a one-parameter family of dispersing (hence hyperbolic, ergodic and mixing) billiards where the correlation function of the collision map decays as $1/n^a$ (here $n$ denotes the discrete time), in which the degree $a \in (1, \infty)$ changes continuously with the parameter of the family, $\beta$. We also derive an explicit relation between the degree $a$ and the family parameter $\beta$.