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 | References | Similar Articles | Additional Information

Abstract: Besicovitch (1934) and Eggleston (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base- 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:
https://doi.org/10.1090/S0002-9947-98-02218-1

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