Résultats de généricité en analyse multifractale

Abstract : The first result involving Hölder regularity and the Baire's categories theorem goes back to 1931. This first notion of genericity supplied by Baire's categories is of a topological nature, and can not permit to understand the size of sets considered. To fill this gap, Christensen defined the measure-theoretic notion of prevalence. There exist also stronger notions of genericity linked with those two first ones. This thesis has two purposes. On one hand, we want to know how smooth are functions in a Sobolev space, for a prevalent set. We show that almost every function is multifractal, as its Hölder exponent changes widely from point to point. We also make the link between the same result in an intersection of Besov spaces and the multifractal formalism given by applications. On the other hand, we compare notions of genericity, testing them with the previous example but also with classical problems from functional analysis.
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Submitted on : Tuesday, April 18, 2006 - 3:33:41 PM
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Aurélia Fraysse. Résultats de généricité en analyse multifractale. Mathématiques [math]. Université Paris XII Val de Marne, 2005. Français. ⟨tel-00012156⟩

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