The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action

First, we give the concepts of G-sequence shadowing property, G-equicontinuity and G-regularly recurrent point. Second, we study their dynamical properties in the inverse limit space under group action. The following results are obtained. (1) The self-mapping ff has the G-sequence shadowing property if and only if the shift mapping σ\sigma has the G¯\overline{G}-sequence shadowing property; (2) The self-mapping ff is G-equicontinuous if and only if the shift mapping σ\sigma is G¯\overline{G}-equicontinuous; (3) RRG¯(σ)=lim←(RRG(f),f)R{R}_{\overline{G}}\left(\sigma )=\underleftarrow{\mathrm{lim}}\left(R{R}_{G}(f),f). These conclusions make up for the lack of theory in the inverse limit space under group action.