CONDITIONAL AND RELATIVE MULTIFRACTAL SPECTRA

In the study of the involved geometry of singular distributions, the u se of fractal and multifractal analysis has shown results of outstandi ng significance. So far, the investigation has focussed on structures produced by one single mechanism which were analyzed with respect to t he ordinary metric or volume. Most prominent examples include self-sim ilar measures and attractors of dynamical systems. In certain cases, t he multifractal spectrum is known explicitly, providing a characteriza tion in terms of the geometrical properties of the singularities of a distribution. Unfortunately, strikingly different measures may possess identical spectra. To overcome this drawback we propose two novel met hods, the conditional and the relative multifractal spectrum, which al low for a direct comparison of two distributions. These notions measur e the extent to which the singularities of two distributions 'correlat e'. Being based on multifractal concepts, however, they go beyond calc ulating correlations. As a particularly useful tool, we develop the mu ltifractal formalism and establish some basic properties of the new no tions. With the simple example of Binomial multifractals, we demonstra te how in the novel approach a distribution mimics a metric different from the usual one. Finally, the applications to real data show how to interpret the spectra in terms of mutual influence of dense and spars e parts of the distributions.