Multifractal Measures

Abstract

Consider a ‘population’ consisting of ‘members’ distributed over a volume of linear size L, that is, over a volume L ^E. The population could, in fact, be the human population distributed over the surface of the earth. The population could also be considered to be the meteorological observation posts, which are unevenly distributed over the globe. The distribution of energy dissipation in space is an example relevant to three-dimensional turbulent flow. The distribution of errors in a transmission line is an example of a one-dimensional population. In physics we routinely consider the distribution of impurities on surfaces and in the bulk. The magnetization of a magnet fluctuates in space. We could consider the local magnetic moments to be members of a population. Many variables fluctuate wildly in space. Gold, for instance, is found in high concentrations at only a few places, in lower concentrations at many places, and in very low concentrations almost everywhere. The point is that this description holds whatever the linear scale is — be it global, on the scale of meters, or on the microscopic scale. Multifractal measures are related to the study of a distribution of physical or other quantities on a geometric support. The support may be an ordinary plane, the surface of a sphere or a volume, or it could itself be a fractal.