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On the absolute continuity of a class of invariant measures

Authors: Tian-You Hu, Ka-Sing Lau and Xiang-Yang Wang
Journal: Proc. Amer. Math. Soc. 130 (2002), 759-767
MSC (2000): Primary 28A80; Secondary 42B10
Published electronically: October 1, 2001
MathSciNet review: 1866031
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Abstract | References | Similar Articles | Additional Information

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

\begin{displaymath}\lambda \mu =\sum_{j=1}^Np_j(\cdot)\mu \circ S_j^{-1}, \end{displaymath}

for some $\lambda>0$. We prove that if the attractor $K$ is an $s$-set and $\mu $ is absolutely continuous with respect to ${\mathcal H}^s\vert _K$, the Hausdorff $s$-dimensional measure restricted on the attractor $K$, then ${\mathcal H}^s\vert _K$ 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.

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Additional Information

Tian-You Hu
Affiliation: Department of Mathematics, University of Wisconsin-Green Bay, Green Bay, Wisconsin 54311

Ka-Sing Lau
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong

Xiang-Yang Wang
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong

Keywords: Absolute continuity, contraction, eigen-function, eigen-measure, iterated function system, singularity
Received by editor(s): September 12, 2000
Published electronically: October 1, 2001
Additional Notes: The first two authors were supported by an HK RGC grant
Communicated by: David Preiss
Article copyright: © Copyright 2001 American Mathematical Society

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