Abstract

To characterize the geometry of a measure, its so-called generalized dimensions d q have been introduced recently. The mathematically precise definition given by Falconer [F2] turns out to be unsatisfactory for reasons of convergence as well as of undesired sensitivity to the particular choice of coordinates in the negative q range. A new definition is introduced, which is based on box-counting too, but which carries relevant information about ¯ also for negative q. In particular, rigorous proofs are provided for the Legendre connection between generalized dimensions and the socalled multifractal spectrum and for the implicit formula giving the generalized dimensions of self-similar measures, which was until now known only for positive q. Fac simile, for personal use only. c fl1995 Academic Press, Inc. 1 Introduction Given a compact set K in Euclidean space IR d , such as the attractor of dynamical systems, the notion of Hausdorff dimension dHD (K) [F2] has been used successfully ...

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