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- W1998422022 abstract "The present paper is the first in a series of three closely related papers in which the inverse measure μ*( dt ) of a given measure μ( dt ) on [0, 1] is introduced. In the first case discussed in detail, μ and μ* 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 spectra of μ( dt ) and μ*( dt ) are shown to be linked by the “inversion formula” f *(α) = α f (1/α). 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 Hölder spectra f H even if the measures μ and μ* 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 Hölder exponents 0 and ∞ 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. The second and third papers in this series make rigorous what is argued with intuitive arguments here. The second paper extends the common scope of the notion of self-similar measures. With this broader class of invariant measures the third paper shows that the multifractal formalism may fail." @default.
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- W1998422022 date "1997-01-01" @default.
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- W1998422022 title "Inverse Measures, the Inversion Formula, and Discontinuous Multifractals" @default.
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- W1998422022 doi "https://doi.org/10.1006/aama.1996.0500" @default.
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