According to the definition of quasiweak almost periodic point, we introduce the definition of QWG(f) in the paper. We studied topological structure of G-quasiweak almost periodic point set. We have the following two results: (1) Let (Y,d′) be a compact G–metric space and f : X → X be equicontinuous. Let {fn}∞n=1 be G-strongly uniform converge to the map f and xk ∈ QWG(fn). If limk → ∞ xk = x, then x ∈ QWG(f); (2) Let (Y,d′) be a compact G-metric space and f : X → X be equicontinuous. If {fn}∞n=1 is G-strongly uniform converge to f, then we have lim sup QWG(fn) ⊂ QWG(f). The conclusions results generalize the corresponding results given in [Journal of Southwest China Normal University: Natural Science Edition 44(2019), 40-44].

We investigated the dynamical relationship between asymptotic average shadow ing property and pointwise Lipschitz shadowing property on the sequence map and the limit map. Then, we have: (1)Suppose {gn } strongly uniformly converge to g · gn has asymptotic average shadowing property implies g has asymptotic average shadowing property. (2) Suppose {gn } strongly uniformly converge to g · gn has fine pointwise shadowing property implies g has pointwise Lipschitz shadowing property. The above results promote the theory development of asymptotic average shadowing property and pointwise Lipschitz shadowing property.

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.