ADVANCES IN APPLIED MATHEMATICS 18 ( 1997 ) Inversion Formula for Continuous

In a previous paper MR] the authors introduced the inverse measure y of a probability measure on 0; 1]. It was argued that the respective multifractal spectra are linked by thèinversion formula' f y () = f(1==). Here, the statements of MR] are put in more mathematical terms and proofs are given for the inversion formula in the case of continuous measures. Thereby, f may stand for the Hausdorr spectrum, the packing spectrum, or the coarse grained spectrum. With a closer look at the special case of self-similar measures we ooer a motivation of the inversion formula as well as a discussion of possible generalizations. Doing so we nd a natural extension of the scope of the notion`self-similar' and a failure of the usual multifractal formalism. c