Chaotic maps in hyperspaces
Suppose \(X\) to be a compact metric space, \(w_1, w_2, \dots , w_m\) invertible contractive selfmaps. The iterated function system \((X;w_1,w_2,\dots ,w_m)\) satisfies the open set condition, if \(X\) is an attractor of the system \((X,w_1,w_2,\dots ,w_m)\) and there is a nonvoid open \(O\subseteq X\) such that \(w_i[O]\cap w_j[O]=\emptyset\) for \(i\neq j\) and \(\bigcup_{1\leq i\leq m} w_i[O]\subseteq O\). The author considers an associated dynamical system \((\mathcal F(X),T)\), where \(\mathcal F(X)\) is the set of all nonempty compact subsets of \(X\) with Hausdorff metric, and the mapping \(T\) is a slight modification of the Barnsley operator: \(T(E)=\bigcup_{1\leq i \leq m} w_i^{-1}[E]\). The paper is devoted to the proof of the following Theorem 1: \((\mathcal F(X),T)\) is a chaotic dynamical system provided that for the initial IFS the open set condition is satisfied. This generalizes the result from [\textit{M. F. Barnsley}, Fractals everywhere (1988; Zbl 0691.58001)].
scientific article; zbMATH DE number 1095860