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 $H.\dim J_{\mathrm{rad}}(f)=\alpha (f)$ 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) $isthecriticalexponentofthePoincar\acute{e}series;andfadmitsauniquenormalizedinvariantdensity7ofdimension$ \delta(f) $.Nowletfbegeometricallyfiniteandsuppose$ f_n \to f $algebraically,preservingcriticalrelations.Whentheconvergenceishorocyclicforeachparabolicpointoff,weshowfnisgeometricallyfinitefor$ n \gg 0 $and$ J(f_n) \to J(f) $intheHausdorfftopology.Iftheconvergenceisradial,theninadditionweshow$ {\rm H.\,dim}\,J(f_{n}) \to {\rm H.\,dim}\,J(f) $.Wegiveexamplesofhorocyclicbutnotradialconvergencewhere$ {\rm H.\,dim}\,J(f_{n}) \to 1 > {\rm H.\,dim}\,J(f) = 1/2 + \epsilon $.WealsogiveasimpledemonstrationofShishikur{a}^{\prime }sresultthatthereexist$ 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.