Polynomial rate of mixing for a family of billiard flows

TL;DR

This paper demonstrates that certain billiard flows exhibit polynomial decay in their correlation functions, by establishing the Holder continuity of the return function on stable and unstable manifolds.

Contribution

It provides a rigorous proof of polynomial mixing rates for specific billiard systems, extending previous understanding of their statistical properties.

Findings

Correlation functions decay polynomially over time

Return function is Holder continuous on stable/unstable manifolds

Supports polynomial mixing behavior in studied billiard flows

Abstract

We prove that the continuous correlation function decrease polynomially for two families of billiard studied by Chernov and Zhang. The main computation is to show that the return function is Holder on stable and unstable manifold.