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
Connectedness of attractors of a certain family of IFSs cover
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Abstract

Let XX be a Banach space and f,g ⁣:XXf,g\colon X\rightarrow X be contractions. We investigate the set

Cf,g:={wX ⁣:theattractorofIFSFw={f,g+w}isconnected.C_{f,g}:=\{w\in X\colon\mathrm {the\: attractor\: of\: IFS} \:\mathcal F_w=\{f,g+w\}\: \mathrm {is\: connected}.

The motivation for our research comes from papers of Miculescu and Mihail, where it was shown that Cf,gC_{f,g} is a countable union of compact sets, provided f,gf,g are linear bounded operators with f,g<1\parallel f\parallel,\parallel g\parallel < 1 and such that ff is compact. Moreover, in the case when XX is finitely dimensional, such sets have been intensively investigated in the last years, especially when ff and gg 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 XX has infinite dimension then Cf,gC_{f,g} is very small set (at least nowhere dense) provided f,gf,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