Conformal measures for rational functions revisited

We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism. Introduction. In this paper we recall from [U2] the notion of conical points and analyze some of its aspects. The idea of conical points has been used implicitly in [DU2], [DU3], [U1], [U3] and other papers of Denker and Urbański. Recently this idea has been used for example in [BMO] to study conformal measures and in [MM] to characterize the Hausdorff dimension and the Poincaré exponent of the Julia sets for certain rational functions. Note that McMullen used the term “radial Julia set” instead of “conical limit set” in analogy with Kleinian groups. We would also like to remark that our approach here is one possible means for examining these notions in the case of parabolic or “geometrically finite” rational maps, that is, those whose Julia sets contain no critical points but some rationally indifferent periodic points. In fact, in these cases (and others also) our construction shows that the h-dimensional Hausdorff measure, where h is the Hausdorff dimension of the Julia set, is supported on the conical set. From this it is not so hard to show that the dimension of the conical set equals the dimension of the measure, hence also equals the Poincaré exponent defined by McMullen and the dimension of the Julia set. 1. Conical points. Let f : C → C be a rational function of degree d ≥ 2. Following [U2], by analogy with the theory of Kleinian groups, we call 1991 Mathematics Subject Classification: 58F08, 28A80, 58F11, 28A78. Research of R. D. Mauldin supported by NSF Grant DMS 9502952. Research of M. Urbański supported by NSF Grant DMS 9502952.