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Let be a metric space and be a continuous map. The notion of the -average shadowing property ( ASP ) for a continuous map on – space is introduced and the relation between the ASP and average shadowing property(ASP)is investigated. We show that if has ASP, then has ASP for every . We prove that if a map be pseudo-equivariant with dense set of periodic points and has the ASP, then is weakly mixing. We also show that if is a –expansive pseudo-equivariant homeomorphism that has the ASP and is topologically mixing, then has a -specification. We obtained that the identity map on has the ASP if and only if the orbit space ⁄ of is totally disconnected. Finally, we show that if is a pseudoequivariant map, and the trajectory map ⁄ is a covering map, then has the ASP if and only if the induced map  ̆ ⁄ ⁄ has ASP.

Some Chaotic Properties of G- Average Shadowing Property