Journal of Statistical Physics, Vol. 113, Nos. 3/4, November 2003, p. 411-430.
Let (S(t), t >/ 0) be a homogeneous fragmentation of ]0, 1[ with no loss of mass. For x from ]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 from ]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.