Markov Processes and Discrete Multifractals

Abstract

Fractals and multifractals are a natural consequence of self-similarity resulting from scale-independent processes. Multifractals are spatially intertwined fractals which can be further grouped into two classes according to the characteristics of their fractal dimension spectra: continuous and discrete multifractals. The concept of multifractals emphasizes spatial associations between fractals and fractal spectra. Distinguishing discrete multifractals from continuous multifractals makes it possible to describe discrete physical processes from a multifractal point of view. It is shown that multiplicative cascade processes can generate continuous multifractals and that Markov processes result in discrete multifractals. The latter result provides not only theoretical evidence for existence of discrete multifractals but also a fundamental model illustrating the general properties of discrete multifractals. Classical prefractal examples are used to show how asymmetrical Markov process can be applied to generate prefractal sets and discrete multifractals. The discrete multifractal model based on Markov processes was applied to a dataset of gold deposits in the Great Basin, Nevada, USA. The gold deposits were regarded as discrete multifractals consisting of three spatially interrelated point sets (small, medium, and large deposits) yielding fractal dimensions of 0.541 for the small deposits ( 500 tons Au), respectively.