Measure Complexity and Rigid Systems

In this paper we introduce two metrics:the max metric d_(n,q)and the mean metric d_(n,q).We give an equivalent characterization of rigid measure preserving systems by the two metrics.It turns out that an invariant measureμon a topological dynamical system(X,T)has bounded complexity with respect to d_(n,q)if and only ifμhas bounded complexity with respect to d_(n,q)if and only if(X,B_X,μ,T)is rigid.We also obtain computation formulas of the measure-theoretic entropy of an ergodic measure preserving system(resp.the topological entropy of a topological dynamical system)by the two metrics dn,q and dn,q.