Remarks on limit sets of infinite iterated function systems

If an iterated function system (IFS) is finite, it is well known that there is a unique non-empty compact invariant set K and that K = π(I∞), where π is the coding map. For an infinite IFS, there are two different sets generalising K, namely π(I∞) and its closure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\pi(I^\infty)}}$$\end{document}. In this paper we investigate the relations between these sets and their Hausdorff dimensions. In particular, we show how to construct an IFS for any pair of prescribed dimensions for π(I∞) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\pi(I^\infty)}\setminus \pi(I^\infty)}$$\end{document} . Moreover, we investigate a set which depends only on the first iteration of an IFS, and characterise its relation to the abovementioned sets. This also extends and clarifies recent results by Mihail and Miculescu, who investigated the coding map for an infinite IFS and a condition for this map to be onto. Finally, we study the special case of one-dimensional IFS and show that in terms of the relations of the abovementioned sets these systems exhibit some very special features which do not generalise to higher dimensional situations.