Great BritainExceptions to the multifractal formalism for discontinuousmeasuresby

In an earlier paper MR] the authors introduced the inverse measure y (dt) of a given measure (dt) on 0; 1] and presented thèinversion formula' f y () = f(1==) which was argued to link the respective multifractal spectra of and y. A second paper RM2] established the formula under the assumption that and y 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 7 ! y creates a new class of discontinuous multifractals. Calculating explicitly we nd that the inversion formula holds only for thè`ne multifractal spectra' and not for thècoarse' ones. As a consequence, the multifractal formalism fails for this class of measures. A natural explanation is found when drawing parallels to equilibrium measures. In the context of our work it becomes natural to consider the degenerate HH older exponents 0 and 1. (fac simile for personal use c Cambridge Philosophical Society)