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

This article concerns a class of open billiards consisting of a finite number of strictly convex, non-eclipsing obstacles K. The non-wandering set M0 of the billiard ball map is a topological Cantor set, and its Hausdorff dimension has been previously estimated for billiards in ℝ2 using well-known techniques. We extend these estimates to billiards in ℝn 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 ℝn 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. © Canadian Mathematical Society 2013.