Sensitivity of iterated function systems

TL;DR

This paper explores the sensitivity and equicontinuity of generalized iterated function systems (IFSs) generated by relations, establishing new conditions for sensitivity and introducing the concept of weak topologically exact IFSs.

Contribution

It generalizes sensitivity and equicontinuity concepts to relation-generated IFSs and introduces weak topologically exact IFSs, providing new criteria for sensitivity.

Findings

Non-minimal weak topologically exact IFSs are sensitive.

Several examples demonstrate sensitivity in non-minimal systems.

Relationships between transitivity, sensitivity, and equicontinuity are clarified.

Abstract

The present work is concerned with the eqiucontinuity and sensitivity of iterated function systems (IFSs). Here, we consider more general case of IFSs, i.e. the IFSs generated by a family of relations. We generalize the concepts of transitivity, sensitivity and equicontinuity to these kinds of systems. This note investigates the relationships between these concepts. Then, several sufficient conditions for sensitivity of IFSs are presented. We introduce the notion of weak topologically exact for IFSs generated by a family of relations. It is proved that non-minimal weak topologically exact IFSs are sensitive. That yields to different examples of non-minimal sensitive systems which are not an M-system. Moreover, some interesting examples are given which provide some facts about the sensitive property of IFSs.