In this work,by virtue of the properties of weakly almost periodic points of a dynamical system (X,T) with at least two points,the authors prove that,if the measure center M(T) of T is the whole space,that is,M(T) =X,then the following statements are equivalent:(1) (X,T) is ergodic mixing; (2) (X,T) is topologically double ergodic;(3) (X,T) is weak mixing; (4) (X,T) is extremely scattering;(5) (X,T) is strong scattering; (6) (X × X,T × T) is strong scattering;(7) (X × X,T × T) is extremely scattering;(8) For any subset S of N with upper density 1,there is a c-dense Fσ-chaotic set with respect to S.As an application,the authors show that,for the sub-shift σA of finite type determined by a k × k-(0,1) matrix A,σA is strong mixing if and only if σA is totally transitive.