Negative fractal dimensions and multifractals

Abstract

A new notion of fractal dimension is defined. When it is positive, it effectively falls back on known definitions. But its motivating virtue is that it can take negative values, which measure usefully the degree of emptiness of empty sets. The main use concerns random multifractals for which f(α) < 0 for some α's. The positive f(α) are show to define a "typical" distribution of the measure, while the negative f(α) rule the sampling variability. Negative dimensions are best investigated using "supersamples." Applications are to turbulence and to DLA. © 1990.