Multifractal spectra of fragmentation processes

Let (S(t), t greater than or equal to 0) be a homogeneous fragmentation of] 0, 1[ with no loss of mass. For x is an element of] 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 v(typ) > 0 such that almost every point x is an element of] 0, 1[has speed v(typ). Nonetheless, for v in a certain range, the random set G(v) of points of speed v, is dense in] 0, 1[, and we compute explicitly the spectrum v --> Dim(G(v)) where Dim is the Hausdorff dimension.