G-Expansibility and G-Almost Periodic Point under Topological Group Action

Abstract

Firstly, the new concepts of $G-$ expansibility, $G-$ almost periodic point, and $G-$ limit shadowing property were introduced according to the concepts of expansibility, almost periodic point, and limit shadowing property in this paper. Secondly, we studied their dynamical relationship between the self-map $f$ and the shift map $\sigma$ in the inverse limit space under topological group action. The following new results are obtained. Let $\left(X,d\right)$ be a metric $G-$ space and $\left(X_f,\overline{G},\overline{d},\sigma \right)$ be the inverse limit space of $\left(X,G,d,f\right)$ . (1) If the map $f:X⟶X$ is an equivalent map, then we have $A{P}_{\overline{G}}\left(\sigma \right)=\underleftarrow{\text{Lim}}\left(A{p}_G\left(f\right),f\right)$ . (2) If the map $f:X⟶X$ is an equivalent surjection, then the self-map $f$ is $G-$ expansive if and only if the shift map $\sigma$ is $\overline{G}-$ expansive. (3) If the map $f:X⟶X$ is an equivalent surjection, then the self-map $f$ has $G-$ limit shadowing property if and only if the shift map $\sigma$ has $\overline{G}-$ limit shadowing property. The conclusions of this paper generalize the corresponding results given in the study by Li, Niu, and Liang and Li . Most importantly, it provided the theoretical basis and scientific foundation for the application of tracking property in computational mathematics and biological mathematics.