Some Chaotic Properties of Average Shadowing Property

Abstract

Let be a metric space and be a continuous map. Thenotion of the -average shadowing property ( ASP ) for a continuous map on –space is introduced and the relation between the ASP and average shadowingproperty(ASP)is investigated. We show that if has ASP, then has ASP forevery . We prove that if a map be pseudo-equivariant with dense set ofperiodic points and has the ASP, then is weakly mixing. We alsoshow that if is a –expansive pseudo-equivariant homeomorphism that has theASP and is topologically mixing, then has a -specification. Weobtained 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 pseudoequivariantmap, and the trajectory map ⁄ is a covering map, thenhas the ASP if and only if the induced map ̆ ⁄ ⁄ has ASP.