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 script G signv, of points of speed v, is dense in ]0, 1[, and we compute explicitly the spectrum v → Dim(script G signv) where Dim is the Hausdorff dimension