An expansion estimate for dispersing planar billiards with corner points

TL;DR

This paper proves that dispersing planar billiards with corner points, no cusps, and bounded horizon satisfy the necessary expansion condition for strong mixing properties, extending known results to more complex billiard geometries.

Contribution

It establishes the expansion condition for dispersing billiards with corner points, which was previously assumed but not proven for such geometries.

Findings

Expansion condition holds for all dispersing billiards with corner points

Strong mixing properties are guaranteed for these billiards

Results extend the class of billiards known to satisfy decay of correlations

Abstract

It is known that the dynamics of planar billiards satisfies strong mixing properties (e.g. exponential decay of correlations) provided that some expansion condition on unstable curves is satisfied. This condition has been shown to always hold for smooth dispersing planar billiards, but it needed to be assumed separately in the case of dispersing planar billiards with corner points. We prove that this expansion condition holds for any dispersing planar billiard with corner points, no cusps and bounded horizon.