Multifractal Spectra of Fragmentation Processes

Let (S(t),t0) 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 v _typ>0 such that almost every point x]0,1 has speed v _typ. Nonetheless, for v in a certain range, the random set _v of points of speed v, is dense in ]0,1, and we compute explicitly the spectrum vDim( _v ) where Dim is the Hausdorff dimension.