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 Jrad(f), and showing that every rational map satisfies { H.\,dim}\,J_{{ rad}}(f) = α(f) where α(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) = δ (f) , where δ(f) is the critical exponent of the Poincaré series; and f admits a unique normalized invariant density 7 of dimension δ(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 fn is geometrically finite for n 0 and J(f_n) → J(f) in the Hausdorff topology. If the convergence is radial, then in addition we show { H.\,dim}\,J(fₙ) → { H.\,dim}\,J(f) . We give examples of horocyclic but not radial convergence where { H.\,dim}\,J(fₙ) → 1 > { H.\,dim}\,J(f) = 1/2 + ε . We also give a simple demonstration of Shishikura's result that there exist f_n(z) = z^2 + c_n with { H.\,dim}\,J(fₙ) → 2 $. The proofs employ a new method that reduces the study of parabolic points to the case of elementary Kleinian groups.