Abstract

The multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that very much like thermodynamic functions, the generalized fractal dimensions Dq and the f(α) singularity spectrum can be readily determined from the scaling behaviour of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of ‘generalized boxes’. We illustrate our theoretical considerations on pedagogical examples, e.g., devil's staircases and fractional Brownian motions. We also report the results of some recent applications of the wavelet transform modulus maxima method to fully developed turbulence data. Then we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic information about the scaling properties of fractal objects. In particular, we show that a dynamical system which leaves invariant such an object can be uncovered from the space-scale arrangement of its wavelet transform modulus maxima. We elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem. This step towards a statistical mechanics of fractals is illustrated on discrete period- doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos.