Hausdorff dimension in graph directed constructions

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

doi:10.1090/S0002-9947-1988-0961615-4

Hausdorff dimension in graph directed constructions
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by R. Daniel Mauldin and S. C. Williams

Trans. Amer. Math. Soc. 309 (1988), 811-829

DOI: https://doi.org/10.1090/S0002-9947-1988-0961615-4

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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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