Topology of connected self-similar tiles in the plane with disconnected interiors
type: Item
instance of: scholarly article
title: Topology of connected self-similar tiles in the plane with disconnected interiors
zbMATH Open document ID: 1077.37019
author: Sze-Man Ngai
author: Tai-Man Tang
published in: Topology and its Applications
publication date: 2005-06-01T00:00:00Z
review text: Let \(\{f_{i}\}_{i=1}^{m}\) be an iterated function system of injective contractions on \(\mathbb{R}^{2}\) satisfying the open set condition, i.e., there exists a nonempty open set \(U\subseteq \mathbb{R}^{2}\) such that \( \bigcup_{i=1}^{m}f_{i}(U)\subseteq U\) and \(f_{i}(U)\cap f_{j}(U)=\emptyset \) for all \(i\neq j\). Assume that the attractor \(T\) of this function system is connected and has nonempty interior \(T^{\circ }\). For this case, the following two theorems are shown: 1. If \(T^{\circ }\) consists of finitely many components, the closure of at least one component is a topological disk and the closure of each component has at most finitely many holes. Moreover, \(\partial T\) is arcwise connected. 2. If \(T^{\circ }\) has infinitely many components, then the closure of each component is a locally connected continuum. Finally, the authors introduce two conditions (``finite tail property'' and ``infinite replication property'') which insure that the closure of each component of \(T^{\circ }\) is a disk. These conditions are in particular satisfied by the Lévy dragon and the Heighway dragon.
reviewed by: Johann Linhart
Mathematics Subject Classification ID: 37B50
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 52C20
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 2173261
zbMATH Keywords: fractal
zbMATH Keywords: iterated function system
zbMATH Keywords: Lévy dragon
MaRDI profile type: Publication
full work available at URL: j.topol.2004.11.009
OpenAlex ID: W2091470825
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DOI: 10.1016/J.TOPOL.2004.11.009
description: scientific article; zbMATH DE number 2173261