HAUSDORFF DIMENSION AND CONFORMAL DYNAMICS, III - COMPUTATION OF DIMENSION

This paper presents an eigenvalue algorithm for accurately computing t he Hausdorff. dimension of limit sets of Kleinian groups and Julia set s of rational maps. The algorithm is applied to Schottky groups, quadr atic polynomials and Blaschke products, yielding both numerical and th eoretical results. Dimension graphs are presented for (a) the family o f Fuchsian groups generated by reflections in 3 symmetric geodesics; ( b) the family of polynomials f(c)(z) = z(2) + c, c is an element of [ - 1, 1/2]; and (c) the family of rational maps f(t)(z) = z/t+ 1/z, t i s an element of (0, 1]. We also calculate H.dim(Lambda) approximate to 1.305688 for the Apollonian gasket, and H.dim(J(f)) approximate to 1. 3934 for Douady's rabbit, where f(z) = z(2) + c satisfies f(3)(0) = 0.