Attractors of iterated function systems and associated graphs
type: Item
instance of: scholarly article
title: Attractors of iterated function systems and associated graphs
zbMATH Open document ID: 1301.28005
author: Dan Dumitru
author: Alexandru Mihail
published in: Kodai Mathematical Journal
publication date: 2014-09-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.kmj/1404393900
review text: The authors are interested in topological properties of the attractor of an intreated function system with possibly infinitely many functions. An iterated function system is a collection of mappings \(\{ f_i\}_{i \in I}\) from a metric space \((X,d)\) into itself. The attractor is a subset \(K \subset X\) such that \(X = \cup_{i \in I} f_i(K)\). It is a natural question whether topological information about \(K\) can be determined from the IFS itself. Connectivity has already been studied [\textit{A. Mihail}, Real Anal. Exch. 34, No. 1, 195--206 (2009; Zbl 1176.28005)]; [\textit{J. Kigami}, Analysis on fractals. Paperback reprint of the hardback edition 2001. Cambridge: Cambridge University Press (2008; Zbl 1143.28005)]; [\textit{M. Yamaguti} et al., Mathematics of fractals. Transl. from the Japanese by Kiki Hudson. Translations of Mathematical Monographs. 167. Providence, RI: American Mathematical Society (AMS). xi, 78 p. \$ 29.00 (1997; Zbl 0888.58030)]. The topological property of interest in this paper is whether or not \(K\) is a dendrite, defined as a connected metric space such that between any two points there is one class of parameterized curves connecting the two points and any two such curves have the same image in \(X\). The main tool to relate the structure of an attractor \(K\) to the IFS itself is via a ``graph of intersections.'' Let \(I\), the index set, and be the vertices of a graph. There is an edge between \(i,j \in I\) if \(f_i(K) \cap f_j(K) \neq \emptyset\). This graph is \(G_1\). Similarly \(G_m\) has \(I^{m}\) as a vertex set and \(\vec{i},\vec{j} \in I^{m}\) are connected by an edge if \(f_{i_m} \circ f_{i_1}(K) \cap f_{j_m} \circ f_{j_1}(K) \neq \emptyset.\) The main result of the paper is the following theorem: { Theorem:} \textit{If all the graphs \(\{G_m\}_{m=1}^{\infty}\) are trees then \(K\) is a dendrite.} Lastly there are several examples of self-similar fractals derived from iterated functions systems which are dendrites including…
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 6341712
zbMATH Keywords: fractals
zbMATH Keywords: iterated functions systems
zbMATH Keywords: dendrites
reviewed by: Benjamin A. Steinhurst
MaRDI profile type: Publication
OpenAlex ID: W2033745325
DOI: 10.2996/KMJ/1404393900
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description: scientific article; zbMATH DE number 6341712