On the absolute continuity of a class of invariant measures

Let X be a compact connected subset of R-d, let S-j, j = 1,..., N, be contractive self-conformal maps on a neighborhood of X, and let {p(j)(x)}(j=1)(N) be a family of positive continuous functions on X. We consider the probability measure mu that satisfies the eigen-equation [GRAPHICS] for some lambda >0. We prove that if the attractor K is an s-set and mu is absolutely continuous with respect to H-s\(K), the Hausdorff s-dimensional measure restricted on the attractor K, then H-s\K is absolutely continuous with respect to mu (i.e., they are equivalent). A special case of the result was considered by Mauldin and Simon (1998). In another direction, we also consider the L-p-property of the Radon-Nikodym derivative of mu and give a condition for which D-mu is unbounded.