  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.
