Hausdorff dimension in graph directed constructions. (English) Zbl 0706.28007

Summary: We introduce the notion of geometric constructions in \({\mathbb{R}}^ m\) 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.


28A78 Hausdorff and packing measures
28A80 Fractals
Full Text: DOI


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