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 | References | Similar Articles | Additional Information

Abstract: Let be a metric space. For a probability measure on a subset of and a Vitali cover of , we introduce the notion of a -Vitali subcover , and compare the Hausdorff measures of with respect to these two collections. As an application, we consider graph directed self-similar measures and in satisfying the open set condition. Using the notion of pointwise local dimension of with respect to , 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.