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

TL;DR

This paper investigates conditions under which graph-directed attractors cannot be represented as standard iterated function system attractors, revealing that most such attractors are unique to their graph-directed construction.

Contribution

It provides algebraic criteria demonstrating that many graph-directed attractors are not realizable as standard IFS attractors, especially when certain circuits do not pass through a vertex.

Findings

Most COSC GD-IFS attractors are not standard IFS attractors.

Algebraic conditions identify when GD-IFS attractors differ from standard IFS attractors.

Existence of GD-IFS attractors not realizable by standard IFSs under specific graph conditions.

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 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 this vertex is also the attractor of a (COSC) standard IFS. In this paper we show the following complementary result. If a directed circuit does not go through a vertex, then there exists a GD-IFS based on the graph such that the attractor associated with this 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.