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

Zhanjiang Ji,José Francisco Gómez Aguilar

doi:10.1155/2021/7326623

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 σ in the inverse limit space under topological group action. The following new results are obtained. Let X , d be a metric G − space and X f , G ¯ , d ¯ , σ be the inverse limit space of X , G , d , f . (1) If the map f : X ⟶ X is an equivalent map, then we have A P G ¯ σ = Lim ← A p G f , f . (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 σ is 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 σ has 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.

Highlights

Let (X, d) be a metric space and f be a continuous map from X to X
(3) If the map f: X ⟶ X is an equivalent surjection, the self-map f has G− limit shadowing property if and only if the shift map σ has G− limit shadowing property. e conclusions of this paper generalize the corresponding results given in the study by Li, Niu, and Liang and Li
In [1], it is proved that the set of almost periodic points of shift map σ is the inverse limit space formed by the self-map f in AP(f)

Summary

Introduction

Let (X, d) be a metric space and f be a continuous map from X to X. In [1], it is proved that the set of almost periodic points of shift map σ is the inverse limit space formed by the self-map f in AP(f). Let (X, d) be a metric G−space, the map f: X ⟶ X be an equivalent map, and (Xf, G, d, σ) be the inverse limit space of (X, G, d, f). We gave the concepts of G−expansive map and G−limit shadowing property according to the concepts of expansive map and limit shadowing property. G−expansibility means expansibility and G−limit shadowing property means limit shadowing property. Eorems 2 and 3, respectively, generalize the corresponding results given in Niu [18] and Liang and Li [10]

G-Almost Periodic Point under Topological Group Action

G-Expansibility under Topological Group Action

G-Limit Shadowing Property under Topological Group Action

Conclusion