On the absolute continuity of a class of invariant measures |
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| Authors: | Tian-You Hu Ka-Sing Lau Xiang-Yang Wang |
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| Affiliation: | Department of Mathematics, University of Wisconsin-Green Bay, Green Bay, Wisconsin 54311 ; Department of Mathematics, The Chinese University of Hong Kong, Hong Kong ; Department of Mathematics, The Chinese University of Hong Kong, Hong Kong |
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| 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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| Keywords: | Absolute continuity contraction eigen-function eigen-measure iterated function system singularity |
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