The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property

Abstract

Abstract In this paper, we introduce the concepts of G G -Lipschitz tracking property and G G -asymptotic tracking property in metric G G -space and obtain the equivalent conditions of G G -asymptotic tracking property in metric G G -space. In addition, we prove that the self-map f f has the G G -Lipschitz tracking property if and only if the shift map σ \sigma has the G ¯ \overline{G} -Lipschitz tracking property in the inverse limit space under the topological group action. These results generalize the corresponding results in [Proc. Amer. Math. Soc. 115 (1992), 573–580].