Browsing by Author "Mandelbrot, Benoit"
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Item Exceptions to the Multifractal Formalism for Discontinuous Measures(1998-01-15) Riedi, Rudolf H.; Mandelbrot, Benoit; Digital Signal Processing (http://dsp.rice.edu/)In an earlier paper the authors introduced the inverse measure µâ (dt) of a given measure µ(dt) on [0,1] and presented the 'inversion formula' fâ (a) = af(1/a) which was argued to link the respective multifractal spectra of µ and µâ . A second paper established the formula under the assumption that µ and µâ 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->µâ 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 of dynamical systems. In the context of our work it becomes natural to consider the degenerate Hölder exponents 0 and infinity.Item Inverse Measures, the Inversion formula, and Discontinuous Multifractals(1997-01-20) Mandelbrot, Benoit; Riedi, Rudolf H.; Digital Signal Processing (http://dsp.rice.edu/)The present paper is part I of a series of three closely related papers in which the inverse measure m' of a given measure m on [0,1] is introduced. In the first case discussed in detail, both these measures are multifractal in the usual sense, that is, both are linearly self-similar and continuous but not differentiable and both are non-zero for every interval of [0,1]. Under these assumptions the Hölder multifractal spectra of the two measures are shown to be linked by the inversion formula f'(a) = a f(1/a) . 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 Hausdorff spectrum even if the measures m and m' are not continuous while it may fail for the spectrum obtained by the Legendre path. This phenomenon goes along with a loss of concavity in the spectrum. Moreover, with the examples discussed it becomes natural to include the degenerate Hölder exponents 0 and infinity in the Hölder spectra. This present paper is the first of three closely related papers on inverse measures, introducing the new notion in a language adopted for the physicist. Parts II and III 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.Item Inversion Formula for Continuous Multifractals(1997-01-20) Riedi, Rudolf H.; Mandelbrot, Benoit; Digital Signal Processing (http://dsp.rice.edu/)In a previous paper the authors introduced the inverse measure µâ of a probability measure µ on [0,1]. It was argued that the respective multifractal spectra are linked by the 'inversion formula' fâ (a) = af(1/a). Here, the statements of Part I are put in more mathematical terms and proofs are given for the inversion formula in the case of continuous measures. Thereby, f may stand for the Hausdorff spectrum, the pacing spectrum, or the coarse grained spectrum. With a closer look at the special case of self-similar measures we offer a motivation of the inversion formula as well as a discussion of possible generalizations. Doing so we find a natural extension of the scope of the notion 'self-similar' and a failure of the usual multifractal formalism.Item Multifractal Formalism for Infinite Multinomial Measures(1995-01-20) Riedi, Rudolf H.; Mandelbrot, Benoit; Digital Signal Processing (http://dsp.rice.edu/)There are strong reasons to believe that the multifractal spectrum of DLA shows anomalies which have been termed left sided. In order to show that this is compatible with strictly multiplicative structures Mandelbrot et al. introduced a one parameter family of multifractal measures invariant under infinitely many linear maps on the real line. Under the assumption that the usual multifractal formalism holds, the authors showed that the multifractal spectrum of these measure is indeed left sided, i.e. they possess arbitrarily large Hölder exponents and the spectrum is increasing over the whole range of these values. Here, it is shown that the multifractal formalism for self-similar measures does indeed hold also in the infinite case, in particular that the singularity exponents D(q) satisfy the usual equation of self-similar measures and that the multifractal spectrum f(a) is the Legendre transform of (q-1)D(q).