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

Firstly, the new concepts of expansibility, almost periodic point, and 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 and the shift map in the inverse limit space under topological group action. The following new results are obtained. Let be a metric space and be the inverse limit space of . (1) If the map is an equivalent map, then we have . (2) If the map is an equivalent surjection, then the self-map is expansive if and only if the shift map is expansive. (3) If the map is an equivalent surjection, then the self-map has limit shadowing property if and only if the shift map has 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.