Inverse Measures, the Inversion formula, and Discontinuous Multifractals
| dc.citation.bibtexName | article | en_US |
| dc.citation.journalTitle | Advances in Applied Mathematics | en_US |
| dc.contributor.author | Mandelbrot, Benoit | en_US |
| dc.contributor.author | Riedi, Rudolf H. | en_US |
| dc.contributor.org | Digital Signal Processing (http://dsp.rice.edu/) | en_US |
| dc.date.accessioned | 2007-10-31T00:52:40Z | en_US |
| dc.date.available | 2007-10-31T00:52:40Z | en_US |
| dc.date.issued | 1997-01-20 | en_US |
| dc.date.modified | 2004-01-22 | en_US |
| dc.date.submitted | 2004-01-14 | en_US |
| dc.description | Journal Paper | en_US |
| dc.description.abstract | 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. | en_US |
| dc.identifier.citation | B. Mandelbrot and R. H. Riedi, "Inverse Measures, the Inversion formula, and Discontinuous Multifractals," <i>Advances in Applied Mathematics,</i> 1997. | en_US |
| dc.identifier.doi | http://dx.doi.org/10.1006/aama.1996.0500 | en_US |
| dc.identifier.uri | https://hdl.handle.net/1911/20081 | en_US |
| dc.language.iso | eng | en_US |
| dc.subject | Temporary | en_US |
| dc.subject.keyword | Temporary | en_US |
| dc.subject.other | Multifractals | en_US |
| dc.title | Inverse Measures, the Inversion formula, and Discontinuous Multifractals | en_US |
| dc.type | Journal article | en_US |
| dc.type.dcmi | Text | en_US |
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