Hausdorff measures on Julia sets of subexpanding rational maps

Let h be the Hausdorff dimension of the Julia set J(R) of a Misiurewicz's rational map R : {Mathematical expression} (subexpanding case). We prove that the h-dimensional Hausdorff measure H _h on J(R) is finite, positive and the only h-conformal measure for R : {Mathematical expression} up to a multiplicative constant. Moreover, we show that there exists a unique R-invariant measure on J(R) equivalent to H _h .