Correlations for pairs of periodic trajectories for open billiards

TL;DR

This paper establishes asymptotic estimates for pairs of closed trajectories in open billiards, extending previous results to higher dimensions and specific billiard classes with Dolgopyat type estimates.

Contribution

It proves two new asymptotic estimates for pairs of closed billiard trajectories, generalizing prior work on negatively curved surfaces to open billiards in multiple dimensions.

Findings

First estimate valid for all open billiards in any dimension.

Second estimate applies to billiards satisfying Dolgopyat type estimates.

Includes all planar open billiards and certain higher-dimensional cases.

Abstract

In this paper we prove two asymptotic estimates for pairs of closed trajectories for open billiards similar to those established by Pollicott and Sharp for closed geodesics on negatively curved compact surfaces. The first of these estimates holds for general open billiards in any dimension. The more intricate second estimate is established for open billiards satisfying the so called Dolgopyat type estimates. This class of billiards includes all open billiards in the plane and open billiards in $\R^N, N \geq 3$ satisfying some additional conditions.