Hausdorff dimension in graph directed constructions

We introduce the notion of geometric constructions in R m { {R}ᵐ} governed by a directed graph G G and by similarity ratios which are labelled with the edges of this graph. For each such construction, we calculate a number α which is the Hausdorff dimension of the object constructed from a realization of the construction. The measure of the object with respect to H α {H^ } is always positive and σ -finite. Whether the H α {H^ } -measure of the object is finite depends on the order structure of the strongly connected components of G G. Some applications are given.