Inverse Measures , the Inversion Formula , andDiscontinuous

The present paper is part I of a series of three closely related papers in which the inverse measure (dt) of a given measure (dt) on 0; 1] is introduced. In the rst case discussed in detail, and are multifractal in the usual sense, that is, both are linearly self-similar and continuous but not diierentiable and both are non{zero for every interval of 0; 1]. Under these assumptions the HH older spectra of (dt) and (dt) are shown to be linked by thèinversion formula' f () = f(1==). The inversion formula is then subjected to several diverse variations, which reveal telling details of interest to the full understanding of multifractals. The inverse of the uniform measure on a Cantor dust leads us to argue that this inversion formula applies to the HH older spectra f H even if the measures and are not continuous while it may fail for the spectrum f L obtained by the Legendre path. This phenomenon goes along with a loss of concavity in the spectrum f H. Moreover, with the examples discussed it becomes natural to include the degenerate HH older exponents 0 and 1 in the HH older spectra. This present paper is the rst of three closely related papers on inverse measures, introducing the new notion in a language adopted for the physicist. Parts II and III RM2, RM3] make rigorous what is argued with intuitive arguments here. Part II extends the common scope of the notion of self-similar measures. With this broader class of invariant measures part III shows that the multifractal formalism may fail. (Facsimile for personal use.) c 1997 Academic Press 1 Heuristic proof of the inversion formula To begin, let us state once again that a multifractal is not a set but a measure. Many multifractals of interest in physics are supported by fractal sets. However, to gain a 1