American Institute of Mathematical Sciences

July  2016, 36(7): 3993-4014. doi: 10.3934/dcds.2016.36.3993

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

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

Received  April 2015 Revised  November 2015 Published  March 2016

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.
Citation: Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993
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