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

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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.

**[B]**R. BOWEN,*Equilibrium states and the ergodic theory of Anosov diffeomorphisms*, Lecture Notes in Math., no.470, Springer Verlag, 1975 MR**56:1364****[F]**K. FALCONER,*The geometry of Fractal sets*, Cambridge University Press, 1985 MR**88d:28001****[FL]**A.H. FAN AND K.S. LAU,*Iterated function system and Ruelle operator*, J. Math. Anal. and Appl.**231**(1999), 319-344 MR**2001a:37013****[HL]**T.Y. HU AND K.S. LAU,*Hausdorff dimension of the level sets of Rademacher series*, Bull. Polish Acad. Sci. Math.**41**(1993), 11-18 CMP**96:16****[LNR]**K.S. LAU, S.M. NGAI AND H. RAO,*Iterated function systems with overlaps and self-similar measures*, J. London Math. Soc. (2)**63**(2001), 99-116. CMP**2001:06****[MS]**R. MAULDIN AND K. SIMON,*The equivalence of some Bernoulli convolutions to Lebesgue measure*, Proc. Amer. Math. Soc.**126**(1998), 2733-2736 MR**98i:26009****[MU]**R. MAULDIN AND M. URBANSKI,*Dimension and measures in infinite function systems*, Proc. London Math. Soc.**73**(1996), 105-154 MR**97c:28020****[PRSS]**Y. PERES, M. RAMS, K. SIMON AND B. SOLOMYAK,*Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets*Proc. Amer. Math. Soc. (to appear)**[PS]**Y. PERES AND B. SOLOMYAK,*Self-similar measures and intersection of Cantor sets*, Trans. Amer. Math. Soc.**350**(1998), 4065-4087 MR**98m:26009****[PSS]**Y. PERES, W. SCHLAG AND B. SOLOMYAK,*Sixty years of Bernoulli convolutions,*Fractals and Stochastics II, (C. Band, S. Graf and M. Zaehle, eds.), Progress in probability**46**, 39-65. Birhauser, 2000 CMP**2001:02****[Q]**A. QUAS,*Non-ergodicity for expanding maps and -measures*, Ergodic Th. Dym. Systems**16**(1996), 531-543 MR**97d:28025**

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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