Weakly Mixing And Chaos

As we all know, the research of complexity is an important subject of topological dynamical system, such as chaotic property, topological transitivity, topological weakly mixing property, topological entropy and so on. Meanwhile, the internal connection among the concepts of system complexity is also a major subject of topological dynamical system. Recent years, more and more researchers extended the complex concepts in dynamical system to topological group, and introduced different kinds of chaos under group. Thus, exploring the relation between complex concepts of the map on the topological group and chaos of group actions is a significant subject. In this paper, firstly we summarize the relationships between weakly mixing and Li-York chaos, weakly mixing and distributional chaos, weakly mixing and distributional chaos in a sequence, weakly mixing and Kato chaos, weakly mixing and Martelli chaos, weakly mixing and R-T chaos. Secondly, we introduce the strong Kato chaos of group actions on the compact metric space. Lastly, we obtain the main result of present paper as follow:Let ? X, d? be a compact metric space without an isolates point. G is a topological commutative group. If the topological dynamical system ? X, G? is weakly mixing, then ? X, G? is strong Kato chaos.