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Geometry of self-similar measures

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1995-09
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Facultad de Ciencias Económicas y Empresariales. Decanato
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Self-similar measures can be obtained by regarding the self similar set generated by a system of similitudes 1J.i = {<Pi}ieM as the probability space associated with an infinite process of Bernouilli trials with state space 1J.i. These measures are concentrated in Besicovitch sets, which are those sets composed oí points with given asymptotic frequencies in their generating similitudes. In this paper we obtain some geometric-size properties of self-similar measures. We generalize the expression of the Hausdorff and packing dimensiona of such measures to the case when M is countable. We give a precise answer to the problem of determining what packing measures are singular viith respect to self-slmilar measures. Both problems are solved by means of a technique which allows us to obtain efficient coverings of balls by cylinder sets. We also show that Besicovitch sets have infinite packing measure in their dimension.
Las medidas autosemejantes pueden obtenerse considerando el conjunto autosemejante generado por un sistema de semejanzas 1J.i = {<Pi}ieM, como el espacio de probabilidad natural asociado a un proceso infinito de ensayos de Bernouilli con espacio de estados 1J.i. Estas medidas están concentradas en los conjuntos de Besicovitch, que son los conjuntos de puntos cuyas secuencias de semejanzas generadoras tienen frecuencias asintóticas fijadas. En este artículo obtenemos algunas propiedades geométricas de las medidas autosemejantes. Por una parte, generalizamos la fórmula para las dimensiones Hausdorff y packing de las medidas autosemejantes al caso en que M es infinito numerable. También damos una clasificación muy precisa de las medidas packing que son singulares respecto a las medidas autosemejantes. Ambos problemas se resuelven mediante una técnica que permite recubrir de manera eficiente bolas mediante cilindros asociados a la construcción geométrica, Demostramos además que los conjuntos de Besicovitch tienen medida packing infinito en su dimensión.
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