2018/12/16 by Strobin, Filip, Swaczyna, Jarosław
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1812.06427
Let X be a Banach space and f,g:X→ X be contractions. We investigate the set Cf,g:=\w∈ X:\m the attractor of IFS \Fw=\f,g+w\\m is connected\. The motivation for our research comes from papers of Mihail and Miculescu, where it was shown that Cf,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 Cf,g is very small set (at least nowhere dense) provided f,g satisfy some natural conditions.