Self-similar Jordan arcs and the graph directed systems of similarities

We study the attractors $\vec\gamma$ of finite graph directed systems $\mathscr S$ of contracting similarities in $\mathbb R^d$ whose components are Jordan arcs. We prove that every self-similar Jordan arc different from a straight line segment may be partitioned into finitely many nonoverlapping subarcs $\delta_j$ each of which also admits a partition into nonoverlapping images of subarcs $\delta_j$ under contracting similarities. A formal description for this property is given by the multizipper construction.