Propriétés d'ubiquité en analyse multifractale et séries aléatoires d'ondelettes à coefficients corrélés

Abstract : The main purpose of this thesis is the description of the size and large intersection properties of sets arising in the multifractal analysis of certain random processes. To this end, we introduce new classes of sets with large intersection which are associated with general gauge functions and we prove, using ubiquity techniques, that these classes contain certain limsup sets. In particular, this enables us to fully describe the size and large intersection properties of sets issued from the classical theory of Diophantine approximation, as the set of points that are well-approximable by rationals or the set of Liouville numbers. We also supply results of the same type when the rational approximates are required to enjoy certain conditions, as the Besicovitch conditions. In addition, our ubiquity techniques allow us to completely describe the size and large intersection properties of the sets coming into play in the multifractal analysis of Lévy processes or certain lacunary wavelet series. We obtain similar results for a new model of random wavelet series whose coefficients are correlated through a tree-indexed Markov chain. In particular, we determine the law of the spectrum of singularities of this model. To perform this study, we analyze a large class of random fractals which generalize the random recursive constructions previously introduced by many authors.
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https://tel.archives-ouvertes.fr/tel-00185375
Contributor : Arnaud Durand <>
Submitted on : Tuesday, November 6, 2007 - 12:21:30 AM
Last modification on : Thursday, January 11, 2018 - 6:12:17 AM
Long-term archiving on : Monday, September 24, 2012 - 2:51:10 PM

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  • HAL Id : tel-00185375, version 1

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Arnaud Durand. Propriétés d'ubiquité en analyse multifractale et séries aléatoires d'ondelettes à coefficients corrélés. Mathématiques [math]. Université Paris XII Val de Marne, 2007. Français. ⟨tel-00185375⟩

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