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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.

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