Attractors of generalized IFSs that are not attractors of IFSs

Mihail and Miculescu introduced the notion of a generalized iterated function system (GIFS in short), and proved that every GIFS generates an attractor. (In our previous paper we gave this notion a more general setting.) In this paper we show that for any m≥2, there exists a Cantor subset of the plane which is an attractor of some GIFS of order m, but is not an attractor of a GIFS of order m−1. In particular, this result shows that there is a subset of the plane which is an attractor of some GIFS, but is not an attractor of an IFS. We also give an example of a Cantor set which is not an attractor of a GIFS.