On the absolute continuity of a class of invariant measures

Abstract:

Let $X$ be a compact connected subset of ${\mathbb R}^d$, 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 \[ \lambda \mu =\sum _{j=1}^Np_j(\cdot )\mu \circ S_j^{-1}, \] for some $\lambda >0$. We prove that if the attractor $K$ is an $s$-set and $\mu$ is absolutely continuous with respect to ${\mathcal H}^s|_K$, the Hausdorff $s$-dimensional measure restricted on the attractor $K$, then ${\mathcal H}^s|_K$ 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.