Attractors of iterated function systems and associated graphs
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 Hata's tree-like set. The graph based analysis of the attractor of an iterated function system adds an interesting complement to graph-directed constructions of non-self-similar fractals in [\textit{R. D. Mauldin} and \textit{S. C. Williams}, Trans. Am. Math. Soc. 309, No. 2, 811--829 (1988; Zbl 0706.28007)].
scientific article; zbMATH DE number 6341712