Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps

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. dim J(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. dim J(rad)(f) = H. dim J(f) = delta (f), where delta (f) is the critical exponent of the Poincare series; and f admits a unique normalized invariant density mu of dimension delta (f). Now let f be geometrically finite and suppose f(n) --> f algebraically, preserving critical relations. When the convergence is horocyclic for each parabolic point of f, we show f(n) is geometrically finite for n much greater than 0 and J(f(n)) --> J(f) in the Hausdorff topology. If the convergence is radial, then in addition we show H. dim J(f(n)) --> H. dim J(f). We give examples of horocyclic but not radial convergence where H. dim J(f(n)) --> 1 > 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 H. dim J(f(n)) --> 2. The proofs employ a new method that reduces the study of parabolic points to the case of elementary Kleinian groups.