Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard

Paul  Wright

doi:10.3934/dcds.2016.36.3993

Article Contents

2016, Volume 36, Issue 7: 3993-4014. Doi: 10.3934/dcds.2016.36.3993

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Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard

Paul Wright^1,

1.
Department of Mathematics and Statistics, The University of Western Australia, Perth

Received: March 31, 2015

Revised: October 31, 2015

Published: May 2017

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Abstract

We consider open billiards in the plane satisfying the no-eclipse condition. We show that the points in the non-wandering set depend differentiably on deformations to the boundary of the billiard. We use Bowen's equation to estimate the Hausdorff dimension of the non-wandering set of the billiard. Finally we show that the Hausdorff dimension depends differentiably on sufficiently smooth deformations to the boundary of the billiard, and estimate the derivative with respect to such deformations.

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Mathematics Subject Classification: Primary: 37C45, 37D50; Secondary: 37D35.

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