Dimensional Characteristics of the Nonwandering Sets of Open Billiards

Abstract

An open billiard is a dynamical system in which a pointlike particle moves at constant speed in an unbounded domain, reflecting off a boundary according to the classical laws of optics [4]. This thesis is an investigation of dimensional characteristics of the nonwandering set of an open billiard in the exterior of three or more strictly convex bodies satisfying Ikawa's no-eclipse condition [5,11]. The billiard map for these systems is an axiom A diffeomorphism with a finite Markov partition. The nonwandering set is a hyperbolic set with stable and unstable manifolds satisfying a certain reflection property. The characteristics we investigate include the topological and measure-theoretic entropy, topological pressure, Lyapunov exponents, lower and upper box dimensions and the Hausdorff dimension of the nonwandering set. In particular, we investigate the dependence of the Hausdorff dimension on deformations to the boundary of the billiard obstacles. While the dependence of dimensional characteristics on perturbations of a system has been studied before [6,9], this is the first time this question has been answered for dynamical billiards.We find upper and lower bounds for the Hausdorff dimension using two different methods: one involving bounding the size of curves on convex fronts and the other using Bowen's equation and the variational principle for topological pressure. Both methods lead to the same upper and lower bounds. In the first method, we use a wellknown recurrence relation for the successive curvatures of convex fronts to find bounds on the size of the fronts. This allows us to construct Lipschitz (but not bi-Lipschitz) homeomorphisms between the nonwandering set and the one-sided symbol space. From there we obtain estimates of the dimension. Kenny [7] used this method for open billiards in the plane. We extend it to higher dimensions and make improvements to the results in the plane. This work has been published in [13].

Differentiability of the Largest Lyapunov Exponent for Non-Planar Open Billiards

“…Let K(α) be a C r,r billiard deformation in R n with r ≥ 3, r ≥ 1. Let V * j and V j be as in (8) and (9). Then, V * j and V j are at least C s , where s = min{r − 2, r } and there exist constants C (s ) n > 0 depending only on n and s such that…”

“…The initial boundary parameterisation and deformation parameter α are used to parameterise the obstacle borders. This was introduced in [9]. Billiard deformations and notations in the subsequent theorems are explained in Section 4.…”

“…, n − 1, for a fixed ξ ∈ Σ α , is differentiable with respect to α and its higher derivative is bounded by a constant independent of α and j. Theorem 5. [9] Let K(α) be a C r,r billiard deformation with r ≥ 2, r ≥ 1. Then for all t = 1, 2, .…”