MULTIFRACTAL SURFACES: LUCENA AND STANLEY APPROACHES

We review multifractal surfaces focusing on a comparison between Lucena and Stanley approaches. The multifractal model as presented by Stanley is basically a multinomial measure over a standard partition of a square. The Lucena approach is geometric oriented, the multifractality is not imposed over a regular lattice, but the lattice itself follows a partition where area tiles obey a multifractal distribution. The non-trivial tilling has a distribution of neighbors of lattice elements that shows a fat tail. Despite the strong differences in the two bidimensional models, both Liacir and Stanley multifractals can be reduced to the same object in one dimension. The message of this article is that there is no unique multifractal object in two dimensions and, as a consequence, we should caution about algorithms that estimate multifractal spectrum in two dimensions because it is not clear what kind of multifractality is being measured. Finally we propose a mixed Lucena-Stanley bimultifractal, an object that combines a multifractal measure over a geometric multifractal tilling.