The research on G-shadowing property and G-periodic shadowing property

The dynamical relationship between G-shadowing property and G-periodic shadowing property is studied in metric G-space. The definitions of G-shadowing property and G-periodic shadowing property are given. By using the properties of equivalent mapping and pseudo equivalent mapping, the following results are obtained:（1） let（ X,d） be compact metric G-space,G be commutative and compact group and f: X → X be an equivariant map. If f has G-periodic shadowing property, then ■ be compact metric G-space, G be compact group and f:X → X be an equivariant map. If f has G-shadowing property and PG（f）= WG（f）, then f has G-periodic shadowing property;（3） let（ X,d） be compact metric G-space, G be commutative and compact group and f:X → X be a pseudo equivariant map. If f is G-expansive map and f has G-shadowing property, then f has G-periodic shadowing property. These results generalize the conclusions of shadowing property and periodic shadowing property in metric spaces.