A topological approach to fuzzy iterated function systems

Abstract

In the paper we unify two extensions of the classical Hutchinson--Barnsley theory - the topological and the fuzzy-set approaches. We show that a fuzzy iterated function system (fuzzy IFS) on a Tychonoff space $X$ which is contracting w.r.t. some admissible multimetric, generates a natural fuzzy attractor in the hyperspace $\mathcal{K}_{\mathcal{F}}(X)$ of all compact fuzzy sets. As a consequence, we prove that a fuzzy IFS on a Hausdorff topological space which is topologically contracting admits a fuzzy attractor in a bit weaker sense. Our discussion involves investigations on topologies on the hyperspace $\mathcal{K}_{\mathcal{F}}(X)$ which are suitable for establishing convergence of sequences of iterations of a fuzzy Hutchinson operator.