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.
