Exceptions to the Multifractal Formalism for Discontinuous Measures

In an earlier paper MR the authors introduced the inverse measure y dt of a given measure dt on and presented the inversion formula f f which was argued to link the respective multifractal spectra of and y A second paper RM 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 y creates a new class of discontinuous multifractals Calculating explicitly we nd that the inversion formula holds only for the ne 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 H older exponents and fac simile for personal use c Cambridge Philosophical Society