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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Hausdorff measures, dimensions and mutual singularity
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by Manav Das PDF
Trans. Amer. Math. Soc. 357 (2005), 4249-4268 Request permission

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.
References
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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
  • MR Author ID: 632693
  • Email: manav@louisville.edu
  • Received by editor(s): May 19, 1997
  • Published electronically: June 13, 2005
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2156710