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.

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In this article, we are trying to see the Iterated function system or more generally a skew product as formation a semigroup action system. We will try to implement more semigroup action properties on itself. For this purpose, we introduce a semigroup action corresponding Iterated function system and explain more properties of itself then with considering Hutchitson operator on Iterated function system, we try consider the corresponding semigroup action for Iterated function system and achive more its properties.