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

DOI:
https://doi.org/10.1090/S0002-9939-01-06363-8

Published electronically:
October 1, 2001

MathSciNet review:
1866031

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a compact connected subset of , let , be contractive self-conformal maps on a neighborhood of , and let be a family of positive continuous functions on . We consider the probability measure that satisfies the eigen-equation

for some . We prove that if the attractor is an -set and is absolutely continuous with respect to , the Hausdorff -dimensional measure restricted on the attractor , then is absolutely continuous with respect to (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 -property of the Radon-Nikodym derivative of and give a condition for which is unbounded.

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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:
https://doi.org/10.1090/S0002-9939-01-06363-8

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