A characterization of topologically transitive attributes for a class of dynamical systems

Abstract

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 F σ-chaotic set with respect to S.

As an application, the authors show that, for the sub-shift σ _A of finite type determined by a k × k-(0, 1) matrix A, σ _A is strong mixing if and only if σ _A is totally transitive.

Keywords

Weakly almost periodic point / Measure center / Topologically transitive attribute / Chaotic set

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