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

  Published:2013-08-10
 Printed: Dec 2013
  • Paul Wright,
    Mathematics Department, University of Western Australia, Western Australia
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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.
Keywords: dynamical systems, billiards, dimension, Hausdorff dynamical systems, billiards, dimension, Hausdorff
MSC Classifications: 37D20, 37D40 show english descriptions Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D20 - Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D40 - Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
 

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