Some Chaotic Properties of G -- Average Shadowing Property.

Let (M,ɗ ) be a metric G-space , ɸ ∶M→M be a continuous map. The notion of the G -average shadowing property (G ASP ) for a continuous map on G -space are introduced and the relation between the G ASP and average shadowing property(ASP)are investigated. We show that if ɸ has GASP, then ɸ^m has GASP for every m∈N. We proved that if a map ɸ be pseudo-equivariant with dense set of G_ɸ-periodic points and has the G ASP then ɸ is weakly G-mixing. We also showed that if ɸ is a G -expansive pseudo-equivariant homeomorphism has the GASP and ɸ is topologically G-mixing then ɸ has a G -specification. We obtained that the identity map ɸ on M has the G ASP if and only if the orbit space M⁄G of M is totally disconnected. Finally, we showed that if ɸ be a pseudo-equivariant map, and the trajectory map, Ψ∶M →M⁄G , is covering map then ɸ has the GASP if and only if the induced map ɸ ̆ ∶M⁄G → M⁄G , has GASP. [ABSTRACT FROM AUTHOR]