Estimates of Hausdorff dimension for non-wandering sets of higher dimensional open billiards

Abstract

This article concerns a class of open billiards consisting of a finite number of strictly convex, non-eclipsing obstacles $K$. The non-wandering set $M_0$ of the billiard ball map is a topological Cantor set and its Hausdorff dimension has been previously estimated for billiards in $\mathbb{R}^2$, using well-known techniques. We extend these estimates to billiards in $\mathbb{R}^n$, and make various refinements to the estimates. These refinements also allow improvements to other results. We also show that in many cases, the non-wandering set is confined to a particular subset of $\mathbb{R}^n$ formed by the convex hull of points determined by period 2 orbits. This allows more accurate bounds on the constants used in estimating Hausdorff dimension.