Multifractal Spectra of Fragmentation Processes

Let (S(t),t≥0) be a homogeneous fragmentation of ]0,1[ with no loss of mass. For x∈]0,1[, we say that the fragmentation speed of x is v if and only if, as time passes, the size of the fragment that contains x decays exponentially with rate v. We show that there is vtyp>0 such that almost every point x∈]0,1[ has speed vtyp. Nonetheless, for v in a certain range, the random set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$G$$ \end{document}v of points of speed v, is dense in ]0,1[, and we compute explicitly the spectrum v→Dim(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$G$$ \end{document}v) where Dim is the Hausdorff dimension.