The thermodynamics of fractals revisited with wavelets

TL;DRAbstract

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

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