# Hausdorff Dimension and Conformal Dynamics II: Geometrically Finite Rational Maps

 Title: Hausdorff Dimension and Conformal Dynamics II: Geometrically Finite Rational Maps Author: McMullen, Curtis T. Citation: McMullen, Curtis T. 2000. Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps. Commentarii Mathematici Helvetici 75(4): 535–593. Revised 2003. Full Text & Related Files: McMullen_HausdorffGeometricFinite.pdf (613.3Kb; PDF) 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 $$J_{rad}(f)$$, and showing that every rational map satisfies $$H. dimJ_{rad}(f) = \alpha(f)$$ 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 $$H. dimJ_{rad}(f) = H. dimJ(f) = \delta(f)$$, where $$\delta(f)$$ is the critical exponent of the Poincar´e series; and $$f$$ admits a unique normalized invariant density $$\mu$$ of dimension $$\delta(f)$$. Now let $$f$$ be geometrically finite and suppose $$f_n \rightarrow f$$ 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 $$J(f_n) \rightarrow J(f)$$ in the Hausdorff topology. If the convergence is radial, then in addition we show $$H. dim J(f_n) \rightarrow H. dimJ(f).$$ We give examples of horocyclic but not radial convergence where $$H. dim J(f_n) \rightarrow 1 > H. dim J(f) = \frac{1}{2} + \epsilon$$. We also give a simple demonstration of Shishikura’s result that there exist $$fn(z) = z^2 + c_n$$ with $$H. dimJ(f_n) \rightarrow 2$$. The proofs employ a new method that reduces the study of parabolic points to the case of elementary Kleinian groups. Published Version: doi:10.1007/s000140050140 Other Sources: http://www.math.harvard.edu/~ctm/papers/index.html Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:3445996 Downloads of this work: