Chaotic maps in hyperspace

Abstract

The dynamical system \((\mathcal{F}(X),T)\) which arises from an iterated function system \((X;w_1,\ldots ,w_m)\), where \(X\) is a compact metric space identified with the attractor of the system and the \(w_i\)’s are contractive invertible maps, is chaotic provided that the iterated function system satisfies the open set condition. The map \(T\) on the hyperspace \(\mathcal{F} (X)\) of the closed subsets of \(X\) is defined for a closed subset \(E\) as \begin{equation*} T(E)=w_1^{-1}(E)\cup \ldots \cup w_m^{-1}(E). \end{equation*} This extends results about the shift dynamical system for the non-overlapping case \cite{ba}.