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Typical Borel measures on $[0,1]d$ satisfy a multifractal formalism

Abstract : In this article, we prove that in the Baire category sense, measures supported by the unit cube of $\R^d$ typically satisfy a multifractal formalism. To achieve this, we compute explicitly the multifractal spectrum of such typical measures $\mu$. This spectrum appears to be linear with slope 1, starting from 0 at exponent 0, ending at dimension $d$ at exponent $d$, and it indeed coincides with the Legendre transform of the $L^q$-spectrum associated with typical measures $\mu$.
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https://hal.archives-ouvertes.fr/hal-00795546
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Submitted on : Thursday, February 28, 2013 - 1:37:24 PM
Last modification on : Monday, April 25, 2022 - 3:03:48 PM

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Zoltán Buczolich, Stéphane Seuret. Typical Borel measures on $[0,1]d$ satisfy a multifractal formalism. Nonlinearity, IOP Publishing, 2010, 23 (11), pp.2905-2918. ⟨10.1088/0951-7715/23/11/010⟩. ⟨hal-00795546⟩

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