Inversion Formula for Continuous Multifractals
Riedi, Rudolf H.
In a previous paper the authors introduced the inverse measure <i>Âµ</i><sup>â </sup> of a probability measure <i>Âµ</i> on [0,1]. It was argued that the respective multifractal spectra are linked by the 'inversion formula' <i>f</i><sup>â </sup>(<i>a</i>) = <i>a</i><i>f</i>(1/<i>a</i>). Here, the statements of Part I are put in more mathematical terms and proofs are given for the inversion formula in the case of continuous measures. Thereby, <i>f</i> may stand for the Hausdorff spectrum, the pacing 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 notion 'self-similar' and a failure of the usual multifractal formalism.