Let be a Banach space and be contractions. We investigate the set
The motivation for our research comes from papers of Miculescu and Mihail, where it was shown that is a countable union of compact sets, provided are linear bounded operators with and such that is compact. Moreover, in the case when is finitely dimensional, such sets have been intensively investigated in the last years, especially when and 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 has infinite dimension then is very small set (at least nowhere dense) provided satisfy some natural conditions.
Cite this article
Filip Strobin, Jaroslaw Swaczyna, Connectedness of attractors of a certain family of IFSs. J. Fractal Geom. 7 (2020), no. 3, pp. 219–231DOI 10.4171/JFG/89