Firstly, we introduce the lower and upper dimensions for a measure defined on a metric space. Secondly, we establish the dimension formulas and characterize the unidimensional measures which were introduced by J.-P. Kahane. Lastly, we give some applications of these to the calculus of dimensions and the multifractal analysis of certain well known measures such as Lebesgue measures on Cantor sets, Gibbs measures, Markov measures and Riesz products etc.
Let T be a d×d matrix with integer entries and with eigenvalues >1 in modulus. Let f be a lipschitzian function of positive order. We prove that the series converges almost everywhere with respect to Lebesgue measure provided that .
On introduit une nouvelle méthode pour démontrer le théorème de Weyl et le théorème de Erdős-Taylor concernant l’équirépartition mod 1 de . Cette méthode fait intervenir des produits de Riesz et s’adapte bien au cas de plusieures dimensions.
We give a simple proof of the sufficiency of a log-lipschitzian condition for the uniqueness of G-measures and g-measures which were studied by G. Brown, A. H. Dooley and M. Keane. In the opposite direction, we show that the lipschitzian condition together with positivity is not sufficient. In the special case where the defining function depends only upon two coordinates, we find a necessary and sufficient condition. The special case of Riesz products is discussed and the Hausdorff dimension of...
En utilisant le théorème de Ruelle d'opérateur de transfert, nous démontrons que la moyenne 2 Σ ||^w|| de la localisation fréquentielle pour les paquets d'ondelettes admet un équivalent de la forme cρ (c > 0, 1 < ρ < √2). Cela améliore une inégalité antérieurement obtenue par Coifman, Meyer et Wickerhauser. Des estimations numériques de ρ sont obtenues pour des filtres de Daubechies.
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