On the decay of fragments in homogeneous fragmentations
We consider a mass-conservative fragmentation of the unit interval. The main purpose of this work is to specify the Hausdorff dimension of the set of locations having exactly an exponential decay. The study relies on an additive martingale which arises naturally in this setting, and a class of Levy process constrained to stay in a finite interval.
Through defining the conceptions of minimal closed chain of q dimensions and fundamental closed chain of q dimensions and index of fundamental closed chain and rank of system of fundamental closed chains and others,the structure of homology group with integral coefficients is described in detail.This description offers an easy method on computing homology group with integral coefficients.\;