Connectedness of Invariant Sets of Graph-Directed IFS

In this paper, we study the connectedness of the invariant sets of a graph-directed iterated function system(IFS). For a graph-directed IFS with N states, we construct N graphs. We prove that all the invariant sets are connected, if and only if all the N graphs are connected; in this case, the invariant sets are all locally connected and path connected. Our result extends the results on the connectedness of the self-similar sets.