Escape Rates and Physically Relevant Measures for Billiards with Small Holes

TL;DR

This paper investigates how small holes in a 2D billiard table affect escape rates and invariant measures, establishing convergence of the limiting distribution to the system's invariant measure as holes shrink.

Contribution

It introduces a framework for analyzing escape rates and conditionally invariant measures in billiards with small holes, extending understanding of open dynamical systems.

Findings

Existence of a common escape rate for a broad class of initial distributions.

The limiting distribution is conditionally invariant and analogous to the SRB measure.

As holes vanish, the limiting distribution converges to the invariant measure of the closed system.

Abstract

We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map.