Abstract
Let (M,É— ) be a metric G-space and ɸ ∶M→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 ɸ has the G Per.SP, then ɸ^n has the G Per.SP for every n∈N. We show that if ɸ is a G –expansive and has the G SP then ɸ has the G Per.SP, and if the map ɸ on compact metric G-space has G-chain transitive and the G Per.SP, then ɸ has the G SP with G-transitivity. We show that the map ɸ on compact metric G-space, ɸ is a G–expansive and G-chain mixing, if ɸ^n has G Per.SP for some n∈N,such that n≠1 then ɸ has G Per.SP. Moreover, we prove that if a map ɸ be pseudo-equivariant with dense set of G_ɸ-periodic points which has the G Per.SP and G-average shadowing property(GASP) then ɸ is G-chain mixing. Finally, we show that if (M,É— ) is a compact metric G-space having two points at least, ɸ be a G-distal homeomorphism and ɸ is G-chain mixing , then ɸ does not have the G Per.SP.

Key words: G-shadowing; periodic shadowing; G-periodic shadowing; G –expansive; The G-average shadowing; G-transitivity ; topologically G-chain mixing.