A Characterization of Topologically Transitive Attributes for a Class of Dynamical Systems

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 Fα-chaotic set with respect to S. As an application, the authors show that, for the sub-shift aA of finite type determined by a k × k-（0, 1） matrix A, erA is strong mixing if and only if aA is totally transitive.