Exceptions to the multifractal formalism for discontinuous measures

In an earlier paper [ MR ] the authors introduced the inverse measure μ [dagger] ( dt ) of a given measure μ( dt ) on [0, 1] and presented the ‘inversion formula’ f [dagger] (α)=α f (1/α) which was argued to link the respective multifractal spectra of μ and μ [dagger] . A second paper [ RM2 ] established the formula under the assumption that μ and μ [dagger] 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 μ[map ]μ [dagger] 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. In the context of our work it becomes natural to consider the degenerate Holder exponents 0 and ∞.