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

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.(H) For distinct 1 ≤ i, j, k ≤ m, the convex hull of K i ∪ K j is disjoint from K k .

Differentiability of the largest Lyapunov exponent for planar open billiards

“…In this paper, we estimate the largest Lyapunov exponent for open billiard in R 2 . We demonstrate that the Lyapunov exponent depends continuously on a parameter α related to a deformation of the billiard as defined in [21]. Moreover, we prove that the Lyapunov exponent is differentiable with respect to the deformation parameter α.…”

“…In this section, we consider some changes to the billiards in the plane, such as moving, rotating, and changing the shape of one or multiple obstacles. This kind of billiard transformation is called a billiard deformation as defined in [21]. We describe this deformation by adding an extra parameter α ∈ [0, b] for some b ∈ R + , which is called the deformation parameter, to the parametrization of the boundary of obstacles i.e., if the boundary of an obstacle parametrized by ϕ(u), it will become ϕ(u, α).…”

“…We describe this deformation by adding an extra parameter α ∈ [0, b] for some b ∈ R + , which is called the deformation parameter, to the parametrization of the boundary of obstacles i.e., if the boundary of an obstacle parametrized by ϕ(u), it will become ϕ(u, α). In this section, we provide the definition a billiard deformation as defined in [21]. In addition, we describe the propagation of unstable manifolds for billiard deformations.…”

“…Definition 4.1. [21] For any α ∈ I = [0, b], let K(α) be a subset of R 2 . For integers r ≥ 4, r ′ ≥ 2, we call K(α) a C r,r ′ -billiard deformation (i.e.…”

Dimensional Characteristics of the Nonwandering Sets of Open Billiards

“…Finally, we show that when the billiard deformation is real analytic, the Hausdorff dimension is also real analytic with respect to α. This work has been submitted for publication (see [14]). For higher dimensions, there is no exact equation for the Hausdorff dimension to differentiate because the billiard ball map is nonconformal.…”