Hausdorr Dimension and Conformal Dynamics Ii: Geometrically Nite Rational Maps

This paper investigates several dynamically de ned dimensions for rational maps f on the Riemann sphere, and gives a systematic development modeled on the theory for Kleinian groups. The radial Julia set is de ned and we show H: dim(Jrad(f)) = (f), the minimal dimension of an f -invariant density. The map f is geometrically nite if every critical point in the Julia set is preperiodic. In this case we show H: dim(Jrad(f)) = H: dim(J(f)) = (f), the critical exponent of the Poincar e series; and f admits a unique normalized invariant density of dimension (f). Now let f be geometrically nite and suppose fn ! f algebraically, preserving critical relations. If the convergence is horocyclic for each parabolic point of f , then fn is geometrically nite for n 0 and J(fn) ! J(f) in the Hausdor topology. If the convergence is radial, then H: dim J(fn)! H: dim J(f). We show there exist fn(z) = z2 + cn such that H: dim J(fn) ! 2, and give other applications to quadratic polynomials. The proofs employ a new method that reduces the study of parabolic points to the case of elementary Kleinian groups.