Dynamical Property of the Shift Map under Group Action

Abstract

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 X , d 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.