Inversion Formula for Continuous Multifractals

Abstract

In a previous paper the authors introduced the inverse measure μ†of a probability measure μ on [0,1]. It was argued that the respective multifractal spectra are linked by the "inversion formula"f†(α)=αf(1/α). Here, the statements of the previous paper are put into more mathematical terms and proofs are given for the inversion formula in the case of continuous measures. Thereby,fmay stand for the Hausdorff spectrum, the packing spectrum, or the coarse grained spectrum. With a closer look at the special case of self-similar measures we offer a motivation of the inversion formula as well as a discussion of possible generalizations. Doing so we find a natural extension of the scope of the notation "self-similar" and a failure of the usual multifractal formalism. © 1997 Academic Press.