Singularity of self-similar measures with respect to Hausdorff measures

Manuel Morán Cabré and Jose María Rey Simó
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Manuel Morán Cabré: Departamento de Análisis Económico y Economía Cuantitativa. Universidad Complutense de Madrid., https://www.ucm.es/fundamentos-analisis-economico2
Jose María Rey Simó: Departamento de Análisis Económico y Economía Cuantitativa. Universidad Complutense de Madrid., https://www.ucm.es/fundamentos-analisis-economico2

No 95-03, Documentos de trabajo de la Facultad de Ciencias Económicas y Empresariales from Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales

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.

Keywords: Self-similar measures; Hausdorff measures. (search for similar items in EconPapers)
Pages: 16 pages
Date: 1995
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