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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Singularity of self-similar measures
with respect to Hausdorff measures


Authors: Manuel Morán and José-Manuel Rey
Journal: Trans. Amer. Math. Soc. 350 (1998), 2297-2310
MSC (1991): Primary 28A78, 28A80
MathSciNet review: 1475691
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Abstract: Besicovitch (1934) and Eggleston (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base-$p$ expansions. We extend this analysis to self-similar sets, by replacing the frequencies of figures with the frequencies of the generating similitudes. We focus on the interplay among such sets, self-similar measures, and Hausdorff measures. We give a fine-tuned classification of the Hausdorff measures according to the singularity of the self-similar measures with respect to those measures. We show that the self-similar measures are concentrated on sets whose frequencies of similitudes obey the Law of the Iterated Logarithm.


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Additional Information

Manuel Morán
Affiliation: Departamento de Análisis Económico, Universidad Complutense, Campus de Somo- saguas, 28223 Madrid. Spain
Email: ececo06@sis.ucm.es

José-Manuel Rey
Affiliation: Departamento de Análisis Económico, Universidad Complutense, Campus de Somo- saguas, 28223 Madrid. Spain
Email: ececo07@sis.ucm.es

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02218-1
PII: S 0002-9947(98)02218-1
Keywords: Self--similarity, Hausdorff measures, dimension function, Law of the Iterated Logarithm.
Received by editor(s): January 17, 1996
Additional Notes: Research partially supported by Ente Público Puertos del Estado
Article copyright: © Copyright 1998 American Mathematical Society