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Singularity of self-similar measures with respect to Hausdorff measures

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Abstract

Besicoviteh (1941) and Egglestone (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.

Suggested Citation

  • Manuel Morán Cabré & Jose María Rey Simó, 1995. "Singularity of self-similar measures with respect to Hausdorff measures," Documentos de trabajo de la Facultad de Ciencias Económicas y Empresariales 95-03, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales.
    • Handle: RePEc:ucm:doctra:95-03
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    Keywords

    Self-similar measures; Hausdorff measures.;

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