Remarks on limit sets of infinite iterated function systems
If an iterated function system (IFS) in \({\mathbb R}^{m}\) is finite, then there exists a unique non-empty compact invariant set \(K\) which can be coded by the full shift space \(\Sigma:= \pi(K)\) over some finite alphabet (where \(\pi\) denotes the coding map). If the IFS is infinite, there are two different sets generalizing \(K\), namely, the set \(\pi(\Sigma)\) and its closure \(\overline{\pi(\Sigma)}\). The paper studies some aspects of the fractal geometry of these two sets. One of the results is that, given any two positive numbers \(s\) and \(t\) less than \(m\), there exists an IFS such that the Hausdorff dimension of \(\pi(\Sigma)\) is equal to \(s\) and the Hausdorff dimension of \(\overline{\pi(\Sigma)}\setminus \pi(\Sigma)\) is equal to \(t\). The paper closes with a study of the one-dimensional case, which turns out to be special in the sense that the sets \(\pi(\Sigma)\) and \(\overline{\pi(\Sigma)}\) have very special features which do not generalize to higher dimensions.
scientific article; zbMATH DE number 6110475