Geometry of self-similar measures

Laburpena

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.