Markov processes and discrete multifractals

Fractals and multifractals are a natural consequence of self-similarity res ulting from scale-independent processes. Multifractals are spatially intert wined fractals which can be further grouped into two classes according to t he characteristics of their fractal dimension spectra: continuous and discr ete multifractals. The concept of multifractals emphasizes spatial associat ions between fractals and fractal spectra. Distinguishing discrete multifra ctals from continuous multifractals makes it possible to describe discrete physical processes from a multifractal point of view. It is shown that mult iplicative cascade processes can generate continuous multifractals and that Markov processes result in discrete multifractals. The latter result provi des not only theoretical evidence for existence of discrete multifractals b ut also a fundamental model illustrating the general properties of discrete multifractals. Classical prefractal examples are used to show how asymmetr ical Markov process can be applied to generate prefractal sets and discrete multifractals. The discrete multifractal model based on Markov processes w as applied to a dataset of gold deposits in the Great Basin, Nevada, USA. T he gold deposits were regarded as discrete multifractals consisting of thre e spatially interrelated point sets (small, medium, and large deposits) yie lding fractal dimensions of 0.541 for the small deposits (<25 tons Au) 0.29 6 for the medium deposits (25-500 tons Au) and 0.09 for the large deposits (>500 tons Au), respectively.