On generalized iterated function systems defined on ℓ∞-sum of a metric space

Abstract

Miculescu and Mihail in 2008 introduced a concept of a generalized iterated function system (GIFS in short), a particular extension of classical IFS. Instead of families of selfmaps of a metric space X, they considered families of mappings defined on finite Cartesian product Xm. It turned out that a great part of the classical Hutchinson–Barnsley theory has natural counterpart in this GIFSs' case. Recently, Secelean extended these considerations to mappings defined on the space ∑∞(X) of all bounded sequences of elements of X and obtained versions of the Hutchinson–Barnsley theorem for appropriate families of such functions. In the paper we study some further aspects of Secelean's setting. In particular, we introduce and investigate a bit more restrictive framework and we show that some problems of the theory have more natural solutions within such a case. Finally, we present an example which shows that this extended theory of GIFSs gives us fractal sets that cannot be obtained by any IFSs or even by any GIFSs.