Projective Cones for Sequential Dispersing Billiards

TL;DR

This paper develops a new mathematical framework using projective cones to analyze the statistical properties of various types of dispersing billiards, including sequential, open, and chaotic scattering systems.

Contribution

It introduces a novel construction of Birkhoff cones for dispersing billiards, enabling the study of complex billiard dynamics beyond traditional regular systems.

Findings

Applicable to sequential billiards with changing collisions

Allows analysis of open billiards with exits or obstacles

Extends to chaotic scattering and Lorentz gas models

Abstract

We construct Birkhoff cones for dispersing billiards, which are contracted by the action of the transfer operator. This construction permits the study of statistical properties not only of regular dispersing billiards but also of sequential billiards (the billiard changes at each collision in a prescribed manner), open billiards (the dynamics exits some region or dies when hitting some obstacle) and many other examples. In particular, we include applications to chaotic scattering and the random Lorentz gas.