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 geodesics; (b) the family of polynomials \(f_c(z) = z^2 +c, c \in [-1, \frac{1}{2}]\); and (c) the family of rational maps \(ft(z) = \frac{z}{t} + \frac{1}{z}, t \in (0, 1]\). We also calculate \(H. dim (\wedge) \approx 1.305688\) for the Apollonian gasket, and \(H. dim (J( f)) \approx 1.3934\) for Douady’s rabbit, where \(f(z) = z^2 + c\) satisfies \(f^3(0) = 0\).