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Estimates of Hausdorff Dimension for the Non-Wandering Set of an Open Planar Billiard

Published online by Cambridge University Press:  20 November 2018

Robert Kenny
Robert Kenny*
Affiliation:
The University of Western Australia, School of Mathematics & Statistics (M019), 35 Stirling Highway, Crawley, WA 6009, Australia e-mail: rkenny@maths.uwa.edu.au
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Abstract

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The billiard flow in the plane has a simple geometric definition; the movement along straight lines of points except where elastic reflections are made with the boundary of the billiard domain. We consider a class of open billiards, where the billiard domain is unbounded, and the boundary is that of a finite number of strictly convex obstacles. We estimate the Hausdorff dimension of the nonwandering set ${{M}_{0}}$ of the discrete time billiard ball map, which is known to be a Cantor set and the largest invariant set. Under certain conditions on the obstacles, we use a well-known coding of ${{M}_{0}}$ [Mor91] and estimates using convex fronts related to the derivative of the billiard ball map [Sto03] to estimate the Hausdorff dimension of local unstable sets. Consideration of the local product structure then yields the desired estimates, which provide asymptotic bounds on the Hausdorff dimension's convergence to zero as the obstacles are separated.

Keywords

37D5037C4528A78
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 1 , 01 February 2004 , pp. 115 - 133
Copyright
Copyright © Canadian Mathematical Society 2004

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