Sums of Dirac masses and conditioned ubiquity

Abstract

Multifractal formalisms hold for certain classes of atomless measures μ obtained as limits of multiplicative processes. This naturally leads us to ask whether non trivial discontinuous measures obey such formalisms. This is the case for a new kind of measures, whose construction combines additive and multiplicative chaos. This class is defined by ν γ , σ = ∑ j ⩾ 1 b − j γ / j 2 ∑ k = 0 b j − 1 μ ( [ k b − j , ( k + 1 ) b − j ) ) σ δ k b − j ( supp ( μ ) = [ 0 , 1 ] , b integer ⩾ 2 , γ ⩾ 0 , σ ⩾ 1 ). Under suitable assumptions on the initial measure μ, ν γ , σ obeys some multifractal formalisms. Its Hausdorff multifractal spectrum h ↦ d ν γ , σ ( h ) is composed of a linear part for h smaller than a critical value h γ , σ , and then of a concave part when h ⩾ h γ , σ . The same properties hold for the Hausdorff spectrum of some function series f γ , σ constructed according to the same scheme as ν γ , σ . These phenomena are the consequences of new results relating ubiquitous systems to the distribution of the mass of μ. To cite this article: J. Barral, S. Seuret, C. R. Acad. Sci. Paris, Ser. I 339 (2004).