Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps

  • Curtis T. McMullen

    Harvard University, Cambridge, USA

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 Jrad(f), and showing that every rational map satisfies where 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 {\rm H.\,dim}\,J_{{\rm rad}}(f) = {\rm H.\,dim}\,J(f) = \delta (f) \delta(f) \delta(f) f_n \to f n \gg 0 J(f_n) \to J(f) {\rm H.\,dim}\,J(f_{n}) \to {\rm H.\,dim}\,J(f) {\rm H.\,dim}\,J(f_{n}) \to 1 > {\rm H.\,dim}\,J(f) = 1/2 + \epsilon f_n(z) = z^2 + c_n {\rm H.\,dim}\,J(f_{n}) \to 2 $. The proofs employ a new method that reduces the study of parabolic points to the case of elementary Kleinian groups.

Cite this article

Curtis T. McMullen, Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps. Comment. Math. Helv. 75 (2000), no. 4, pp. 535–593

DOI 10.1007/S000140050140