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

Wright, Paul

Canadian journal of mathematics, Tome 65 (2013) no. 6, pp. 1384-1400

Résumé

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.

Détail

DOI : 10.4153/CJM-2013-030-0

Mots-clés : 37D20, 37D40, dynamical systems, billiards, dimension, Hausdorff

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