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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.

##### MSC:
 28A78 Hausdorff and packing measures 28A80 Fractals
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##### References:
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