Sums of Dirac masses and conditioned ubiquity

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 ${\nu }_{\gamma \text{,}\sigma }={\sum }_{j?1}{b}^{?j\gamma }/j^2{\sum }_{k=0}^{b^j?1}\mu (k{b}^{?j}\text{,}{(k+1)b^{?j}))}^{\sigma }{\delta }_{k{b}^{?j}}$ ($\text{supp}(\mu )=0\text{,}1]\text{,}b$ integer $?2\text{,}\gamma ?0\text{,}\sigma ?1$). Under suitable assumptions on the initial measure μ, ${\nu }_{\gamma \text{,}\sigma }$ obeys some multifractal formalisms. Its Hausdorff multifractal spectrum $h?d_{{\nu }_{\gamma \text{,}\sigma }}(h)$ is composed of a linear part for h smaller than a critical value $h_{\gamma \text{,}\sigma }$, and then of a concave part when $h?h_{\gamma \text{,}\sigma }$. The same properties hold for the Hausdorff spectrum of some function series $f_{\gamma \text{,}\sigma }$ constructed according to the same scheme as ${\nu }_{\gamma \text{,}\sigma }$. 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).