Remarks on limit sets of infinite iterated function systems
type: Item
instance of: scholarly article
title: Remarks on limit sets of infinite iterated function systems
zbMATH Open document ID: 1272.28006
author: Martial R. Hille
published in: Monatshefte für Mathematik
publication date: 2012-11-27T00:00:00Z
review text: 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.
reviewed by: Bernd O. Stratmann
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 6110475
zbMATH Keywords: iterated function system
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: invariant set
zbMATH Keywords: limit set
MaRDI profile type: Publication
full work available at URL: s00605-011-0357-6
OpenAlex ID: W2050602086
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DOI: 10.1007/S00605-011-0357-6
description: scientific article; zbMATH DE number 6110475