On Hausdorff measures and SBR measures for parabolic rational maps

If J is the Julia set of a parabolic rational map having Hausdorff dimensio n h < 1, we show that Sullivan's h-conformal measure on J is either absolut ely continuous or orthogonal with respect to the Hausdorff measures defined by the function f(t) = t(h)(log 1/t)(tau), according to whether tau > tau( 0) or tau less than or equal to tau(0) for some explicitly computable tau(0 ) > 0.