The Research of G-Mixing and G-Equicontinuity on the Hyperspace of Topological Group Action

G-mixing and G-equicontinuity have an important significance in terms of theory and application. According to the definition of mixing and equicontinuity, we give the concept of G-mixing and G-equicontinuity in this paper. By inference, some conclusions of the hyperspace were extended to the hyperspace of topological group action. We have the following result: (1)Let (K,H, f̅,G) be the hyperspace of (X, d, G, f) and f : X → X be a pseudo equivariant map. All non-empty open set are invariable for G . If f is a G-mixing map, then f̅ is a G-mixing map; (2)Let (K,H, f̅,G) be the hyperspace of (X,d,G, f) and f : X → X be a equivariant map. Then f is a G-mild mixing map if and only if f̅ is a G-mild mixing map;(3)Let (K,H, f̅,G) be the hyperspace of (X,d,G, f) and f :X → X be a equivariant map. Then f is a G-equicontinuity map if and only if f̅ is a Gequicontinuity map. These results enriched the theory of the G-mixing and G-equicontinuity on the hyperspace of topological group action.