Mixing rate for semi-dispersing billiards with non-compact cusps

TL;DR

This paper investigates the mixing properties of billiard tables with non-compact cusps, demonstrating polynomial mixing speed using a specific infinite measure framework, which was previously unexplored.

Contribution

It introduces a new analysis of mixing for billiards with non-compact cusps under infinite measure, establishing polynomial decay of correlations.

Findings

The billiard with a non-compact cusp is proven to be mixing.

The mixing speed for this billiard is polynomial.

This work applies Krengel and Sucheston's definition of mixing to infinite measure systems.

Abstract

Since the seminal work of Sinai one studies chaotic properties of planar billiards tables. Among them is the study of decay of correlations for these tables. There are examples in the literature of tables with exponential and even polynomial decay. However, until now nothing is known about mixing properties for billiard tables with non-compact cusps. There is no consensual definition of mixing for systems with infinite invariant measure. In this paper we study geometric and ergodic properties of billiard tables with a non-compact cusp. The goal of this text is, using the definition of mixing proposed by Krengel and Sucheston for systems with invariant infinite measure, to show that the billiard whose table is constituted by the x-axis and and the portion in the plane below the graph of $f(x)=\frac{1}{x+1}$ is mixing and the speed of mixing is polynomial.