# Commentarii Mathematici Helvetici

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**Volume 75, Issue 4, 2000, pp. 535–593**

**DOI: 10.1007/s000140050140**

Published online: 2000-12-31

Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps

Curtis T. McMullen^{[1]}(1) 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 $ {\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 PoincarĂ© 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.

*Keywords: *Complex dynamics, iterated rational maps, Julia sets, Hausdorff dimension

McMullen Curtis: Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps. *Comment. Math. Helv.* 75 (2000), 535-593. doi: 10.1007/s000140050140