Some Chaotic Properties of G- Average Shadowing Property

Abstract

 Let  be a metric space and  be a continuous map. The notion of the  -average shadowing property ( ASP )  for a continuous map on  â€“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 pseudo-equivariant map, and  the trajectory  map  is a covering map, then  has the ASP  if and only if the induced map   has ASP.