Zero-dimensional compact metrizable spaces as attractors of generalized iterated function systems (Q2422568)

A generalized iterated function system (GIFS in short) consists of maps defined on a finite Cartesian $m$-th power $X^m$ with values in X (in such a case we say that the GIFS is of order $m$). It turned out that a great part of the classical Hutchinson theory for IFS has a natural counterpart in this GIFS' framework. On the other hand, only few examples of fractal sets are known which are generated by GIFS, but are not IFS's attractors. In this paper, the authors study $0$-dimensional compact metrizable spaces from the perspective of GIFS theory. They prove that each such space $X$ is homeomorphic to the attractor of some GIFS on the real line. Moreover, they prove that $X$ can be embedded into the real line ${\mathbb R}$ as the attractor of some GIFS of order $m$ and (in the same time) a nonattractor of any GIFS of order $m-1$, as well as it can be embedded as a nonattractor of any GIFS. Then they show that a relatively simple modifications of $X$ delivers spaces whose connected component is ``big'' and which are GIFS attractors not homeomorphic with IFS's attractors. Finally, they show that a generic compact subset of a Hilbert space is not the attractor of any Banach GIFS.