Decay of correlations for billiards with flat points I: channel effect

TL;DR

This paper investigates the decay of correlations in semi-dispersing billiards with flat points, demonstrating polynomial decay rates in both billiard systems and Lorenz gases with infinite horizon.

Contribution

It constructs a specific family of semi-dispersing billiards with flat points and analyzes their mixing rates, revealing polynomial decay of correlations.

Findings

Correlation functions decay polynomially in the studied systems.

The presence of flat points creates a channel effect influencing mixing rates.

The results connect billiard dynamics with Lorenz gas behavior.

Abstract

In this paper we constructed a special family of semidispersing billiards bounded on a rectangle with a few dispersing scatters. We assume there exists a pair of flat points (with zero curvature) on the boundary of these scatters, whose tangent lines form a channel in the billiard table that is perpendicular to the vertical sides of the rectangle. The billiard can be induced to a Lorenz gas with infinite horizon when replacing the rectangle by a torus. We study the mixing rates of the one-parameter family of the semi-dispersing billiards and the Lorenz gas on a torus; and show that the correlation functions of both maps decay polynomially.