Dynamical Property of the Shift Map under Group Action

Firstly, we introduced the concept of $G‐$ Lipschitz tracking property, $G‐$ asymptotic average tracking property, and $G‐$ periodic tracking property. Secondly, we studied their dynamical properties and topological structure and obtained the following conclusions: (1) let $\left(X,d\right)$ be compact metric $G‐$ space and the metric $d$ be invariant to $G$ . Then, $\sigma$ has $\overline{G}‐$ asymptotic average tracking property; (2) let $\left(X,d\right)$ be compact metric $G‐$ space and the metric $d$ be invariant to $G$ . Then, $\sigma$ has $\overline{G}‐$ Lipschitz tracking property; (3) let $\left(X,d\right)$ be compact metric $G‐$ space and the metric $d$ be invariant to $G$ . Then, $\sigma$ has $\overline{G}‐$ periodic tracking property. The above results make up for the lack of theory of $G‐$ Lipschitz tracking property, $G‐$ asymptotic average tracking property, and $G‐$ periodic tracking property in infinite product space under group action.