On the absolute continuity of a class of invariant measures

Abstract

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 {pj(x)} N j=1 be a family of positive continuous functions on X. We consider the probability measure μ that satisfies the eigen-equation λμ= Σ N j=1 pj(.)μ o S -1 j , for some λ > 0. We prove that if the attractor K is an s-set and μ 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 μ (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 μ and give a condition for which D μ is unbounded.