On the Connectivity of the Attractors of Recurrent Iterated Function Systems

Iterated function systems (IFSs) were conceived in the present form by John Hutchinson [4] and popularized by Michael Barnsley [1] and are one of the most common and most general ways to generate fractals. Many of the important examples of sets and functions with special and unusual properties in analysis turn out to be fractal sets or functions whose graph are a fractal sets and a great part of them are attractors of IFSs. There is a current effort to extend the classical Hutchinson’s framework to more general spaces and infinite iterated function systems (IIFSs) or more generally to multifunction systems and to study them. A recent such extension of the IFS theory can be found in [7], where the Lipscomb’s space-which was an important example in dimension theory – can be obtained as an attractor of an IIFS defined in a very general setting. In this setting the attractor can be a closed and bounded set in contrast with the classical theory where only compact sets are considered. Although the fractal sets are defined with measure theory – being sets with noninteger Hausdorff dimension [2], [3] – it turns out that they have interesting topological properties as we can see from the above example [7]. One of the most important result in this direction is given in Theorem 1.2 below (see [11] for a proof) which states when the attractor of an IFS is a connected set. We intend to extend this result to recurrent iterated function system (see [8], [9]; see [5], [6] for a generalization of results from [8]).