A Class of Multinomial Multifractal Measures with Negative (Latent) Values for the “Dimension” f(α)

Abstract

As is well known, fractals are sets of points that possess is the property of being invariant by dilation. When a fractal set is exactly self-similar, or is self-similar in a statistical sense, a central role is played by a positive quantity called fractal dimension, which need not be an integer, and which generalizes the “ordinary” dimension.

In this paper, a further generalization of dimension is introduced and motivated. When its value is positive, it effectively falls back on known definitions of fractal dimension. But its motivating virtue is that its value can be negative, in which case it quantifies and measures usefully the loose idea of “degree of emptiness” of an empty set.

Self-similar multifractals are also geometric objects invariant by dilation, but they are not sets. They are measures, a notion well illustrated by the concrete distributions of probability, of mass or of turbulent dissipation. They are described by a function f(α).

This paper shows that negative dimensions are needed to investigate the statistical properties of certain random self-similar multifractals, namely those for which f(α) < 0 for some α’s, called latent (= present, but hidden). The positive f(α)’s define a “typical” distribution for the measure, while the negative f(α)’s rule the variability between different samples from the same ensemble or population. Moments whose order lies beyond certain thresholds q* and q*min are called latent. Their sample values are extremely sample dependent. Negative dimensions are best investigated using “supersamples.”

In addition to negative f’s, the multifractals investigated here involve a critical exponent q _bottom < 0, such that population moments of exponent q less than q _bottom are infinite. The corresponding sample moments are extremely “ill-behaved.”

The “ordinary” multifractals, whose simplest example is the binomial, are called “manifest” by the author. Compared to them, the multifractals studied in this paper exhibit two “anomalies” that play a central role in the approach to multifractals the author has been developing since 1968. Their study requires nothing but elementary manipulations, making them the simplest examples of the anomalies in question.

Currently, the most active applications of negative dimensions are to the distribution of the dissipation of turbulence and to the distribution of the hitting probability along the boundary of a DLA cluster, which is an example of harmonic measure.