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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hausdorff dimension in graph directed constructions

Authors: R. Daniel Mauldin and S. C. Williams
Journal: Trans. Amer. Math. Soc. 309 (1988), 811-829
MSC: Primary 28A75
MathSciNet review: 961615
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Abstract: 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.

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Keywords: Spectral radius, entropy, Hausdroff measure
Article copyright: © Copyright 1988 American Mathematical Society