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 ðX, dÞ be compact metric G‐space and the metric d be invariant to G. Then, σ has G�‐asymptotic average tracking property; (2) let (X, d) be compact metric G‐space and the metric d be invariant to G. Then, σ has G�‐Lipschitz tracking property; (3) let (X, d) be compact metric G‐space and the metric d be invariant to G. Then, σ has 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.