The Research of G-Almost Periodic Point

Abstract

Firstly, it is introduced that the concepts of G-almost periodic point and G-sequence shadowing property. Then, we discuss the dynamical relationship between sequence map {gk}∞k=1 and limit map g under G-strongly uniform convergence of topological group action. We can get that (1) Let sequence map {gk}∞k=1 be G-strongly uniform converge to the map g where g is equicontinuous and the point sequence {yk}∞k=1 be the G-almost periodic point of sequence map {gk}∞k=1. If limk → ∞ yk = y, then the point y is an G-almost periodic point of the map g; (2) If sequence map {gk}∞k=1 are G-strongly uniform converge to the map g where g is equicontinuous, then limsup APG(gk) ⊂ APG(g); (3) Let sequence map {gk}∞k=1 be G-strongly uniform converge to the map g. If every map gk has G-fine sequence shadowing property, the map g has G-sequence shadowing property. These results generalize the corresponding results given in Ji and Zhang [1] and make up for the lack of theory under G-strongly uniform convergence of group action.