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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
DOI: https://doi.org/10.1090/S0002-9947-1988-0961615-4
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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DOI: https://doi.org/10.1090/S0002-9947-1988-0961615-4
Keywords: Spectral radius, entropy, Hausdroff measure
Article copyright: © Copyright 1988 American Mathematical Society