# Connectedness of attractors of a certain family of IFSs

### Filip Strobin

Lodz University of Technology, Łódź, Poland### Jaroslaw Swaczyna

Lodz University of Technology, Łódź, Poland

## Abstract

Let $X$ be a Banach space and $f,g\colon X\rightarrow X$ 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.

## Cite this article

Filip Strobin, Jaroslaw Swaczyna, Connectedness of attractors of a certain family of IFSs. J. Fractal Geom. 7 (2020), no. 3, pp. 219–231

DOI 10.4171/JFG/89