Ergodic Theory and Dynamical Systems

Research Article

Geometric measures for parabolic rational maps

M. Denkera1 and M. Urbańskia2

a1 Institut für Mathematische Stochastik, Lotzestr. 13, 3400 Göttingen, Germany

a2 Instytut Matematyki UMK, ul. Chopina 12/18, 87–100 Toruń, Poland

Abstract

Let h denote the Hausdorff dimension of the Julia set J(T) of a parabolic rational map T. In this paper we prove that (after normalisation) the h-conformal measure on J(T) equals the h-dimensional Hausdorff measure Hh on J(T), if h ≥ 1, and equals the h-dimensional packing measure Πh on J(T), if h ≤ 1. Moreover, if h < 1, then Hh = 0 and, if h > 1, then Πh(J(T)) = ∞.

(Received November 14 1990)

(Revised February 26 1991)