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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the parametrization of self-similar and other fractal sets


Authors: Miguel Angel Martín and Pertti Mattila
Journal: Proc. Amer. Math. Soc. 128 (2000), 2641-2648
MSC (2000): Primary 28A75, 28A80
Published electronically: March 1, 2000
MathSciNet review: 1664402
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Abstract:

We prove for many self-similar, and some more general, sets $E \subset \mathbb{R}^{n}$ that if $s$ is the Hausdorff dimension of $E$ and $f: \mathbb{R}^{m} \to \mathbb{R}^{n}$ is Hölder continuous with exponent $m/s$, then the $s$-dimensional Hausdorff measure of $E \cap f(\mathbb{R}^{m})$ is $0$.


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Additional Information

Miguel Angel Martín
Affiliation: Departamento de Matemática Aplicada, E.T.S.I. Agrónomos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
Email: mamartin@mat.etsia.upm.es

Pertti Mattila
Affiliation: Department of Mathematics, University of Jyväskylä, P. O. Box 35, FIN-40351 Jyväskylä, Finland
Email: pmattila@math.jyu.fi

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05354-5
PII: S 0002-9939(00)05354-5
Received by editor(s): June 4, 1998
Received by editor(s) in revised form: October 20, 1998
Published electronically: March 1, 2000
Additional Notes: This work was partially done while P. Mattila was visiting the Centre de Recerca Matemàtica in Barcelona supported by the Ministerio de Educacion y Cultura.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society