Connectedness of attractors of a certain family of IFSs

Abstract

Let $X$ be a Banach space and $f,g:X\to X$ be contractions. We investigate the set

The motivation for our research comes from papers of Miculescu and Mihail, where it was shown that $C_{f,g}$ is a countable union of compact sets, provided $f,g$ are linear bounded operators with $\parallel f\parallel ,\parallel g\parallel <1$ and such that $f$ is compact. Moreover, in the case when $X$ is finitely dimensional, such sets have been intensively investigated in the last years, especially when $f$ and $g$ are affine maps. As we will be mostly interested in infinite dimensional spaces, our results can be also viewed as a next step into extending of such studies into infinite dimensional setting. In particular, unlike in the finitely dimensional case, if $X$ has infinite dimension then $C_{f,g}$ is very small set (at least nowhere dense) provided $f,g$ satisfy some natural conditions.