The Hutchinson-Barnsley theory for infinite iterated function systems

Abstract

We show that some results of the Hutchinson-Barnsley theory for finite iterated function systems can be carried over to the infinite case. Namely, if { F i : i ∈ ℕ} is a family of Matkowski's contractions on a complete metric space ( X, d ) such that ( F i x 0 )i∈N is bounded for some x 0 ∈ X , then there exists a non-empty bounded and separable set K which is invariant with respect to this family, that is, . Moreover, given σ ∈ ℕ ℕ and x ∈ X , the limit exists and does not depend on x . We also study separately the case in which ( X, d ) is Menger convex or compact. Finally, we answer a question posed by Máté concerning a finite iterated function system { F 1 ,…, F N } with the property that each of F i has a contractive fixed point.