Parameterized IFS with the asymptotic average shadowing property

TL;DR

This paper extends the concept of asymptotic average shadowing to parameterized iterated function systems (IFS), establishing key properties and relationships, including conditions for chain transitivity and examples differentiating shadowing from asymptotic average shadowing.

Contribution

It introduces the asymptotic average shadowing property for parameterized IFS and proves new theorems, including its relation to chain transitivity and examples illustrating differences from classical shadowing.

Findings

Uniformly contracting IFS have the asymptotic average shadowing property.

Asymptotic average shadowing implies chain transitivity for surjective IFS.

Existence of IFS with asymptotic average shadowing but not shadowing property.

Abstract

In this paper we generalize the notion of asymptotic average shadowing property to parameterized IFS and prove some related theorems on this notion. Specially, this 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 X has the asymptotic average shadowing property then F is chain transitive. Moreover, we introduce some examples and investigate the relationship between the original asymptotic average shadowing property and asymptotic average shadowing property for IFS. For example, there is an IFS F such that has the asymptotic average shadowing property but does not satisfy the shadowing property.