BEYOND CLASSICAL MULTIFRACTAL ANALYSIS USING WAVELETS: UNCOVERING A MULTIPLICATIVE PROCESS HIDDEN IN THE GEOMETRICAL COMPLEXITY OF DIFFUSION LIMITED AGGREGATES

Abstract

We emphasize the wavelet transform as a very promising tool for solving the inverse fractal problem. We show that a dynamical system which leaves invariant a fractal object can be Uncovered from the space-scale arrangement of its wavelet transform modulus maxima. We illustrate our theoretical considerations on pedagogical examples including Bernoulli invariant measures of linear and nonlinear expanding Markov maps as well as the invariant measure of period-doubling dynamical systems at the onset of chaos. We apply this wavelet based technique to analyze the fractal properties of DLA azimuthal Cantor sets defined by intersecting the inner frozen region of large mass off-lattice DLA clusters with a circle. This study clearly reveals the existence of an underlying multiplicative process that is likely to account for the Fibonacci structural ordering recently discovered in the apparently disordered arborescent DLA morphology. The statistical relevance of the golden mean arithmetic to the fractal hierarchy of the DLA azimuthal Cantor sets is demonstrated.