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¶\(\),¶where \( \delta(f) \) is the critical exponent of the Poincare series; and f admits a unique normalized invariant density μ 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.