Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps

Abstract

This paper investigates several dynamically defined dimensions for rational maps f on the Riemann sphere, providing a systematic treatment modeled on the theory for Kleinian groups. We begin by defining the radial Julia set Jrad(f), and showing that every rational map satisfies ${\rm H.\,dim}\,J_{{\rm rad}}(f) = \alpha(f)$ where $\alpha(f)$ is the minimal dimension of an f-invariant conformal density on the sphere. A rational map f is geometrically finite if every critical point in the Julia set is preperiodic. In this case we show {\rm H.\,dim}\,J_{{\rm rad}}(f) = {\rm H.\,dim}\,J(f) = \delta (f) $, where$ \delta(f) $is the critical exponent of the Poincaré series; and f admits a unique normalized invariant density 7 of dimension$ \delta(f) $. Now let f be geometrically finite and suppose$ f_n \to f $algebraically, preserving critical relations. When the convergence is horocyclic for each parabolic point of f, we show fn is geometrically finite for$ n \gg 0 $and$ J(f_n) \to J(f) $in the Hausdorff topology. If the convergence is radial, then in addition we show$ {\rm H.\,dim}\,J(f_{n}) \to {\rm H.\,dim}\,J(f) $. We give examples of horocyclic but not radial convergence where$ {\rm H.\,dim}\,J(f_{n}) \to 1 > {\rm H.\,dim}\,J(f) = 1/2 + \epsilon $. We also give a simple demonstration of Shishikura's result that there exist$ f_n(z) = z^2 + c_n $with$ {\rm H.\,dim}\,J(f_{n}) \to 2 $. The proofs employ a new method that reduces the study of parabolic points to the case of elementary Kleinian groups.