Singularity of self-similar measures with respect to Hausdorff measures

• Autores: Manuel Moran Cabre, José Manuel Rey Simó
• Localización: Documentos de trabajo de la Facultad de Ciencias Económicas y Empresariales, ISSN-e 2255-5471, Nº. 3, 1995, 16 págs.
• Idioma: inglés
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• Resumen

  • 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