Riedi, Rudolf H.Mandelbrot, Benoit2007-10-312007-10-311998-01-152004-01-14R. H. Riedi and B. Mandelbrot, "Exceptions to the Multifractal Formalism for Discontinuous Measures," <i>Mathematical Proceedings Cambridge Philosophical Society,</i> 1998.https://hdl.handle.net/1911/20270Journal PaperIn an earlier paper the authors introduced the <i>inverse measure</i> <i>µ</i>^â(<i>dt</i>) of a given measure <i>µ</i>(<i>dt</i>) on [0,1] and presented the 'inversion formula' <i>f</i>^â(<i>a</i>) = <i>a</i><i>f</i>(1/<i>a</i>) which was argued to link the respective multifractal spectra of <i>µ</i> and <i>µ</i>^â. A second paper established the formula under the assumption that <i>µ</i> and <i>µ</i>^â 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>µ</i>-><i>µ</i>^â 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 Hölder exponents 0 and infinity.engTemporaryMultifractalsJournal articleTemporary