The Strong G-shadowing Property of the Inverse Limit Spaces and the Product Spaces of Group Action

The concept of the strong G-shadowing property is given in the metric spaces under the action of topologi-cal group. Then the dynamical properties of the strong G-shadowing property in the inverse limit spaces and the product spaces under the action of topological group are studied. The following conclusions are obtained. Let the system (X_f, G, d, σ) be the inverse limit spaces of the system (X, G, d, f). Then f has the G-shadowing property if and only if σ has the G-shadowing property. The product map f₁×f₂ has the strong G-shadowing property if and only if the map f₁ has the strong G₁-shadowing property and the map f₂ has the strong G₂-shadowing property. These results enrich the theory of strong G-shadowing property in the inverse limit spaces and the product spaces under the action of topological group.