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

Mauldin, R. Daniel; Williams, S. C.

Trans. Am. Math. Soc. 309, No. 2, 811-829 (1988).

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

Keywords:

geometric graph directed construction in \({\mathbb{R}}^ m\); Hausdorff measure; spectral radius; entropy; similarity ratios; Hausdorff dimension

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