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Estimates of Hausdorff Dimension for Non-wandering Sets of Higher Dimensional Open Billiards

Published online by Cambridge University Press:  20 November 2018

Paul Wright
Paul Wright*
Affiliation:
Mathematics Department, University of Western Australia, Perth, Western Australia, e-mail: paul.e.wright@uwa.edu.au
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Abstract

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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.

Keywords

37D2037D40dynamical systemsbilliardsdimensionHausdorff
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 6 , 01 December 2013 , pp. 1384 - 1400
Copyright
Copyright © Canadian Mathematical Society 2013

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