Exceptions to the Multifractal Formalism for Discontinuous Measures
Riedi, Rudolf H.
In an earlier paper the authors introduced the <i>inverse measure</i> <i>Âµ</i><sup>â </sup>(<i>dt</i>) of a given measure <i>Âµ</i>(<i>dt</i>) on [0,1] and presented the 'inversion formula' <i>f</i><sup>â </sup>(<i>a</i>) = <i>a</i><i>f</i>(1/<i>a</i>) which was argued to link the respective multifractal spectra of <i>Âµ</i> and <i>Âµ</i><sup>â </sup>. A second paper established the formula under the assumption that <i>Âµ</i> and <i>Âµ</i><sup>â </sup> are continuous measures. Here, we investigate the general case which reveals telling details of interest to the full understanding of multifractals. Subjecting self-similar measures to the operation <i>Âµ</i>-><i>Âµ</i><sup>â </sup> creates a new class of discontinuous multifractals. Calculating explicitly we find that the inversion formula holds only for the 'fine multifractal spectra' and not for the 'coarse' ones. As a consequence, the multifractal formalism fails for this class of measures. A natural explanation is found when drawing parallels to equilibrium measures of dynamical systems. In the context of our work it becomes natural to consider the degenerate HÃ¶lder exponents 0 and infinity.