Hausdorff dimension in graph directed constructions

Source abstract

We introduce the notion of geometric constructions in R m {{\mathbf {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 \alpha 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 {\mathcal {H}^\alpha } is always positive and σ \sigma -finite. Whether the H α {\mathcal {H}^\alpha } -measure of the object is finite depends on the order structure of the strongly connected components of G G . Some applications are given.