Graph Iterated Function Systems and Fractal Tops

Abstract

Following the work of Louisa and Michael Barnsley on results in tops of iterated function systems, we extend their work to graph-directed iterated function systems by investigating the relationship between top addresses and shift spaces. For the simplest overlapping interval IFS, we find a sufficient condition for the closure of its tops code space to be a shift space of finite type. Likewise, we find that shift invariance properties do not directly extend to the graph-directed setting.