Let (M,É— ) be a metric G-space and É¸ âˆ¶Mâ†’M be a continuous map. This paper aims to study the idea of the G-periodic shadowing property (G Per.SP ) for a continuous map on G-space and achieves the relative of the G Per.SP with G-shadowing property (G SP). Also, if É¸ has the G Per.SP, then É¸^n has the G Per.SP for every nâˆˆN. We show that if É¸ is a G –expansive and has the G SP then É¸ has the G Per.SP, and if the map É¸ on compact metric G-space has G-chain transitive and the G Per.SP, then É¸ has the G SP with G-transitivity. We show that the map É¸ on compact metric G-space, É¸ is a G–expansive and G-chain mixing, if É¸^n has G Per.SP for some nâˆˆN,such that nâ‰ 1 then É¸ has G Per.SP. Moreover, we prove that if a map É¸ be pseudo-equivariant with dense set of G_É¸-periodic points which has the G Per.SP and G-average shadowing property(GASP) then É¸ is G-chain mixing. Finally, we show that if (M,É— ) is a compact metric G-space having two points at least, É¸ be a G-distal homeomorphism and É¸ is G-chain mixing , then É¸ does not have the G Per.SP.
G-shadowing; periodic shadowing; G-periodic shadowing; G –expansive; The G-average shadowing; G-transitivity ; topologically G-chain mixing.