Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps

  • Curtis T. McMullen

    Harvard University, Cambridge, USA


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.dimJrad(f)=α(f){\rm H.\,dim}\,J_{{\rm rad}}(f) = \alpha(f) where α(f)\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, where \delta(f) isthecriticalexponentofthePoincareˊseries;andfadmitsauniquenormalizedinvariantdensity7ofdimensionis the critical exponent of the Poincaré series; and f admits a unique normalized invariant density 7 of dimension \delta(f) .Nowletfbegeometricallyfiniteandsuppose. Now let f be geometrically finite and suppose f_n \to f algebraically,preservingcriticalrelations.Whentheconvergenceishorocyclicforeachparabolicpointoff,weshowfnisgeometricallyfiniteforalgebraically, preserving critical relations. When the convergence is horocyclic for each parabolic point of f, we show fn is geometrically finite for n \gg 0 andand J(f_n) \to J(f) intheHausdorfftopology.Iftheconvergenceisradial,theninadditionweshowin 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) .Wegiveexamplesofhorocyclicbutnotradialconvergencewhere. 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 .WealsogiveasimpledemonstrationofShishikurasresultthatthereexist. We also give a simple demonstration of Shishikura's result that there exist f_n(z) = z^2 + c_n withwith {\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.

Cite this article

Curtis T. McMullen, Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps. Comment. Math. Helv. 75 (2000), no. 4, pp. 535–593

DOI 10.1007/S000140050140