Abstract: The relationship among G-chain recurrent point, G-nonwandering point, G-limit point and G-recurrent point were studied under the condition that the map f is G-equicontinuous. The following conclusions are obtained: (1)R_G(f)=W_G(f)=Ω_G(f); (2)W_G(f)=CR_G(f)=\bigcap\limits_n=1^\infty f^n(X); (3)f is G-equicontinuous if and only if all point in W_G(f) are G-equicontinuous points. The above conclusions enrich the theory of G-chain recurrent point, G-nonwandering point, G-limit point, and G-recurrent point in metric G-space.