Chaotic property for non-autonomous iterated function system

In this paper, the new concept of non-autonomous iterated function system is introduced and also shown that non-autonomous iterated function system IFS(f_(1,∞)^0,f_(1,∞)^1) is topologically transitive for the metric space of X whenever the system has average shadowing property and its minimal points on X are dense. Moreover, such a system is topologically transitive, whenever, there is a point like z∈U for each open and invariant set U from X so that N(z,U) has a positive upper density. It is also shown that topological transitivity is result of properties of shadowing and chain transitivity. The relation between average shadowing property , topological transitivity and chaotic non-autonomous iterated function system is studied .Moreover, it is also demonstrated that the first two conditions for the definition of chaos results the third condition. The topological mixing of such a system is obtained from shadowing property and chain mixing. Finally, we evaluated that the dynamical system (X, f) has Li-York e chaos under special conditions.