Retracted: Chain Recurrent Point and Devaney Chaos of Topological Groups

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Chain recurrent point, strong chain recurrent point and Devaney chaos have an important significance in terms of theory and application. In this paper, we will introduce the concept of chain recurrent point, strong chain recurrent point and Devaney chaos. By means of properties of the product space and topological conjugation, We study the dynamical properties of chain recurrent point, strong chain recurrent point and Devaney chaos. It is gived that the following conclusions. (1) The set $CR(G)$ and $SCR(G)$ are a closed set. (2) The set $CR(G)$ and $SCR(G)$ are strongly invariable for $G$ . (3) If $G_{1}$ and $G_{2}$ are to be topologically conjugate, then the group $G_{1}$ is Devaney chaotic if and only if the group $G_{2}$ is Devaney chaotic; (4) The group $G_{1}\times G_{2}$ is Devaney chaotic if and only if the group $G_{1}$ and $G_{2}$ are Devaney chaotic. These results enrich the theory of chain recurrent point, strong chain recurrent point and Devaney chaos under the action of topological group