GEOMETRIC MEASURES FOR PARABOLIC RATIONAL MAPS

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 H(h) on J(T), if h greater-than-or-equal-to 1, and equals the h-dimensional packing measure PI(h) on J(T), if h less-than-or-equal-to 1. Moreover, if h < 1, then H(h), = 0 and, if h > 1, then PI(h)(J(T)) = infinity.