Inverse Measures, the Inversion Formula, and Discontinuous Multifractals

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 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 di erentiable and both are non zero for every interval of Under these assumptions the H older spectra of dt and dt are shown to be linked by the inversion formula f f 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 H older spectra fH even if the measures and are not continuous while it may fail for the spectrum fL obtained by the Legendre path This phenomenon goes along with a loss of concavity in the spectrum fH Moreover with the examples discussed it becomes natural to include the degenerate H older exponents and in the H 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 RM RM 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 Academic Press 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 E mail riedi math yale edu riedi rice edu full intuitive understanding of the notion of multifractal unencumbered by extraneous complication relative to its support is best achieved in terms of a measure supported by the interval One begins by de ning the measure dt for the closed intervals of the form t in other words by giving a positive non decreasing function t M t For other intervals is de ned via s t M t M s s t M t M s etc When M t has a derivative M t the measure of an in nitesimal interval t t dt is the ordinary di erential dM t M t dt and has the density M t When M t is discontinuous at t dM t is the value of that discontinuity M t M t In addition M is right continuous Conversely any right continuous non decreasing functionM withM M de nes a measure as above De nition of the inverse of a basic multifractal The usual multifractals are measures that are continuous but not di erentiable In a rst stage we require in addition that M t is strictly increasing so that every interval of t s however small has a non vanishing measure This is equivalent to saying that the measure is supported on the whole interval In a widely used notation it means that D In this case the function M t has a well de ned inverse function M that is right continuous and non decreasing hence de nes a second measure d More precisely denoting the length of an interval I s t by jIj t s we have I M t M s jM I j M I t s jIj Picking a point t at random on with respect to the measure amounts to taking at random on with uniform probability and then taking for t the value M Picking a point at random on with the measure amounts to taking t at random on with uniform probability and then taking for the value M t Heuristic argument for the inversion formula Given a multifractal described by f let us show that the function f of the measure is given by the inversion formula f f First note that a point t of H older exponent corresponds to a point M t of H older exponent lim dt ftg log dt log jdtj lim M dt f g log jM dt j log M dt where the limit is taken over all intervals dt shrinking down to ftg Now divide the interval on the t axis into small intervals of length By the de nition of f EXAMPLES AND COMMENTS the set K of H older exponent can be covered by N f intervals The measure of each of these intervals is approximately In other words the function M t maps these intervals to N intervals each of length covering the set K of points with H older exponent The dimension of this set is therefore