Topology of connected self-similar tiles in the plane with disconnected interiors
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.
scientific article; zbMATH DE number 2173261