Parameterized IFS with the Asymptotic Average Shadowing Property

Abstract

In this paper, we generalize the notion of the asymptotic average shadowing property to parameterized IFS's and prove some related theorems on this notion. Specially, it is proved that every uniformly contracting IFS has the asymptotic average shadowing property. As an important result, we show that if a continuous surjective IFS F on a compact metric space has the asymptotic average shadowing property then F is chain transitive. Moreover, we give some examples to illustrate our approach and compare the asymptotic average shadowing property for IFS with the asymptotic average shadowing property in discrete dynamical systems. For example, we will show that there is an IFS F which has the asymptotic average shadowing property but does not satisfy the shadowing property.