Dual Multifractal Spectra

The multifractal formalism characterizes the scaling properties of a physical density rho as a function of the distance L. To each singularity alpha of the field is attributed a fractal dimension for its support f(alpha). An alternative representation has been proposed by Jensen considering the distribution of distances associated to a fixed mass. Computing these spectra for a multifractal Cantor set, it is shown that these two approaches are dual to each other, and that both spectra as well as the moment scaling exponents are simply related. We apply the same inversion formalism to exponents obtained for turbulent statistics in the GOY shell model and observe that the same duality relation holds here.