New "anomalous" multiplicative multifractals: Left sided ƒ(α) and the modelling of DLA

Abstract

A sharp distinction is drawn between general multiplicative multifractals, as originally introduced by the author, and the more familiar but more restricted class defined by Frisch and Parisi and by Halsey et al. All the general multiplicative multifractals are exactly renormalizable, by design. However (this is important and perhaps suprising), it is shown that the scaling relations that serve to define the restricted class fail to extend to certain multiplicative multifractals. As a result, in addition to the familiar restricted class, the general multifractals are shown to allow for a wide variety of "anomalous" behaviors, several of which are described. We believe that suitable examples of these anomalies are of the most important concrete applications to fully developed turbulence and to diffusion limited aggregates (DLA). This paper concerns several new classes of multifractals we have recently proposed as tools to model the anomalies of DLA. They are "left sided", that is, characterized by αmax=∝ with f{hook}(αmax)=D0. Related multifractals characterized by αmax=0, and even by αmax=∝ and αmin=0, are encountered along the way. © 1990.