ON THE EXISTENCE OF WELL DEFINED SINGULARITIES IN MULTIFRACTALS

Abstract

The two most commonly used interpretations of multifractals are based on two approaches: (i) constructing a histogram of the corresponding probability distribution, and (ii) the study of the spatial distribution of the local singularities characterizing the fractal measures. In this paper we shall consider these approaches with the purpose of investigating some of the open problems arising from their relationship. In particular, we address the question of whether there exists a well defined exponent α describing the strength of the local singularity of a fractal measure. From theoretical and numerical studies of a multiplicative measure defined on the unit interval, we conclude that an a exists for all points whose decimal digits have an asymptotic distribution, and that α(x) depends only on the asymptotic ratio of these digits. We have also examined the connections between the two different definitions of the f(α) spectra.