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 \(J_{rad}(f)\), 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.