On the absolute continuity of a class of invariant measures
Authors:
TianYou Hu, KaSing Lau and XiangYang Wang
Journal:
Proc. Amer. Math. Soc. 130 (2002), 759767
MSC (2000):
Primary 28A80; Secondary 42B10
Published electronically:
October 1, 2001
MathSciNet review:
1866031
Fulltext PDF Free Access
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Abstract: Let be a compact connected subset of , let , be contractive selfconformal maps on a neighborhood of , and let be a family of positive continuous functions on . We consider the probability measure that satisfies the eigenequation
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 RadonNikodym derivative of and give a condition for which is unbounded.
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Additional Information
TianYou Hu
Affiliation:
Department of Mathematics, University of WisconsinGreen Bay, Green Bay, Wisconsin 54311
Email:
HUT@uwgb.edu
KaSing Lau
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
Email:
kslau@math.cuhk.edu.hk
XiangYang 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/S0002993901063638
PII:
S 00029939(01)063638
Keywords:
Absolute continuity,
contraction,
eigenfunction,
eigenmeasure,
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
