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Transactions of the American Mathematical Society

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Hausdorff measures, dimensions and mutual singularity


Author: Manav Das
Journal: Trans. Amer. Math. Soc. 357 (2005), 4249-4268
MSC (2000): Primary 28A78; Secondary 28A80, 60A10
DOI: https://doi.org/10.1090/S0002-9947-05-04031-6
Published electronically: June 13, 2005
MathSciNet review: 2156710
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Abstract: Let $(X,d)$ be a metric space. For a probability measure $\mu$ on a subset $E$of $X$ and a Vitali cover $V$ of $E$, we introduce the notion of a $b_{\mu}$-Vitali subcover $V_{\mu}$, and compare the Hausdorff measures of $E$with respect to these two collections. As an application, we consider graph directed self-similar measures $\mu$ and $\nu$ in $\mathbb{R}^{d}$ satisfying the open set condition. Using the notion of pointwise local dimension of $\mu$with respect to $\nu$, we show how the Hausdorff dimension of some general multifractal sets may be computed using an appropriate stochastic process. As another application, we show that Olsen's multifractal Hausdorff measures are mutually singular.


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Additional Information

Manav Das
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Address at time of publication: Department of Mathematics, University of Louisville, Louisville, Kentucky 40292
Email: manav@louisville.edu

DOI: https://doi.org/10.1090/S0002-9947-05-04031-6
Keywords: Hausdorff measure, Vitali cover, multifractal, (strong) open set condition, stochastic process, stoppings
Received by editor(s): May 19, 1997
Published electronically: June 13, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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