Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF Free Access

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 28A80, 42B10

Retrieve articles in all journals with MSC (2000): 28A80, 42B10


Additional Information

Tian-You Hu
Affiliation: Department of Mathematics, University of Wisconsin-Green Bay, Green Bay, Wisconsin 54311
Email: HUT@uwgb.edu

Ka-Sing Lau
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
Email: kslau@math.cuhk.edu.hk

Xiang-Yang Wang
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
Email: xywang@math.cuhk.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06363-8
PII: S 0002-9939(01)06363-8
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