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.
Highlights
The dimension theory of dynamical systems studies the dimensional characteristics of the invariant sets of dynamical systems
Past work has examined how dimensional characteristics of various dynamical systems can change with respect to perturbations of the system; for example the differentiability of entropy of Anosov flows [11], SRB measures in hyperbolic flows [22], and Hausdorff dimension of horseshoes [14]
The Hausdorff dimension of the non-wandering set has been estimated for open billiards in the plane [13] and in higher dimensions [28]
Summary
The dimension theory of dynamical systems studies the dimensional characteristics (such as Hausdorff dimension) of the invariant sets of dynamical systems. Past work has examined how dimensional characteristics of various dynamical systems can change with respect to perturbations of the system; for example the differentiability of entropy of Anosov flows [11], SRB measures in hyperbolic flows [22], and Hausdorff dimension of horseshoes [14] This kind of problem has not been considered in the context of open billiard systems. In this paper we show that the Hausdorff dimension of the non-wandering set for an open billiard in the plane depends smoothly on perturbations to the boundary of the billiard. The Hausdorff dimension of the non-wandering set was estimated in [13] by investigating convex fronts This was later improved and extended to higher dimensional billiards in [28]. ≤ C, where C is a constant depending only on simple geometrical characteristics of the obstacles
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