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

TL;DR

This paper extends and refines estimates of the Hausdorff dimension for non-wandering sets in higher-dimensional open billiards with convex obstacles, providing more precise bounds and insights into the structure of these sets.

Contribution

It generalizes previous 2D estimates to higher dimensions and introduces refinements that improve the accuracy of Hausdorff dimension bounds for non-wandering sets.

Findings

Hausdorff dimension estimates extended to $ ext{R}^n$

Refinements lead to more accurate bounds

Non-wandering sets often confined to convex hulls of period 2 orbits

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.