On single-matrix graph-directed iterated function systems

We study graph-directed function systems where each contraction in the system has the form f e ( x ) = A − 1 ( x + d e ) , where A is an expanding matrix. We show that a certain discreteness implies the open set condition, and the latter implies the strong open set condition. Hausdorff measures and dimensions (w.r.t. a weak norm) of the invariant sets are investigated. The stationary Markov measures of the system are proved to be translation invariant.