Graph-Directed Iterated Function Systems Satisfying the Generalized Finite Type Condition

We extend the generalized finite type condition to graph-directed iterated function systems with overlaps. Under this condition, we can compute the Hausdorff dimension of the attractor F in terms of the spectral radius of certain weighted incidence matrix. Moreover, if the Haudorff dimension of F is α, then the α-dimensional Hausdorff and packing measures of F are shown to be strictly positive. By assuming in addition that the graph is strongly connected, we show that the Hausdorff, packing, and box dimensions are equal and the a-dimensional Hausdorff and packing measures are finite.