Hausdorff Dimension in Graph Directed Constructions
R. Daniel Mauldin, S. C. Williams · Transactions of the American Mathematical Society · 1988
We introduce the notion of geometric constructions in ${{\mathbf {R}}^m}$ governed by a directed graph $G$ and by similarity ratios which are labelled with the edges of this graph. For each such construction, we calculate a number $\alpha$ which is the Hausdorff dimension of the object constructed from a realization of the construction. The measure of the object with respect to ${\mathcal {H}^\alpha }$ is always positive and $\sigma$-finite. Whether the ${\mathcal {H}^\alpha }$-measure of the object is finite depends on the order structure of the strongly connected components of $G$. Some applications are given.