# Connectedness of attractors of a certain family of IFSs

## TL;DR

This paper studies the set of parameters in an infinite-dimensional Banach space for which the attractor of a certain IFS is connected, extending previous finite-dimensional results and showing that this set is very small under natural conditions.

## Contribution

It extends the analysis of connectedness of IFS attractors from finite-dimensional to infinite-dimensional Banach spaces, revealing the set's smallness under natural conditions.

## Key findings

- In infinite-dimensional spaces, the set of parameters with connected attractors is nowhere dense.
- The results generalize previous finite-dimensional findings to a broader infinite-dimensional context.
- Under certain conditions, the set of parameters with connected attractors is very small (meager).

## Abstract

Let $X$ be a Banach space and $f,g:X\rightarrow X$ be contractions. We investigate the set $$ C_{f,g}:=\{w\in X:\m{ the attractor of IFS }\F_w=\{f,g+w\}\m{ is connected}\}. $$ The motivation for our research comes from papers of Mihail and Miculescu, where it was shown that $C_{f,g}$ is a countable union of compact sets, provided $f,g$ are linear bounded operators with $\pa f\pa,\pa g\pa<1$ 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.