Hausdorff dimension and conformal dynamics, III: Computation of dimension

Abstract

This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials and Blaschke products, yielding both numerical and theoretical results. Dimension graphs are presented for (a) the family of Fuchsian groups generated by reflections in 3 symmetric geodesies; (b) the family of polynomials fc(z) = z2+c, c ∈ [-1, 1/2]; and (c) the family of rational maps ft(z) = z/t + 1/z, t ∈ (0, 1]. We also calculate H.dim (Λ) ≈ 1.305688 for the Apollonian gasket, and H.dim (J(f)) ≈ 1.3934 for Douady's rabbit, where f(z) = z2 + c satisfies f3(0) = 0.