A dichotomy on the self-similarity of graph-directed attractors

Abstract

This paper seeks conditions that ensure that the attractor of a graph directed iterated function system (GD-IFS) cannot be realised as the attractor of a standard iterated function system (IFS). For a strongly connected directed graph, it is known that, if all directed circuits go through a particular vertex, then for any GD-IFS of similarities on $\mathbb{R}$ based on the graph and satisfying the convex open set condition (COSC), its attractor associated with that vertex is also the attractor of a (COSC) standard IFS. In this paper we show the following complementary result. If there exists a directed circuit which does not go through a certain vertex, then there exists a GD-IFS based on the graph such that the attractor associated with that vertex is not the attractor of any standard IFS of similarities. Indeed, we give algebraic conditions for such GD-IFS attractors not to be attractors of standard IFSs, and thus show that ‘almost-all’ COSC GD-IFSs based on the graph have attractors associated with this vertex that are not the attractors of any COSC standard IFS.