• DOI:10.1090/S0002-9947-1988-0961615-4
• Corpus ID: 53967994

Hausdorff dimension in graph directed constructions

@article{Mauldin1988HausdorffDI,
  title={Hausdorff dimension in graph directed constructions},
  author={R. Daniel Mauldin and S. C. Williams},
  journal={Transactions of the American Mathematical Society},
  year={1988},
  volume={309},
  pages={811-829}
}

• R. Mauldin, S. C. Williams
• Published 1 February 1988
• Mathematics
• Transactions of the American Mathematical Society

We introduce the notion of geometric constructions in Rm 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 a which is the Hausdorff dimension of the object constructed from a realization of the construction. The measure of the object with respect to fl"* is always positive and <r-finite. Whether the ^""-measure of the object is finite depends on the order structure of the strongly connected components of G. Some applications are given. A geometric graph directed construction in Rm consists of (1) a finite sequence of nonoverlapping, compact subsets of Rm: J\,...,Jn such that each J, has a nonempty interior, (2) a directed graph G with vertex set consisting of the integers l,...,n and similarity maps Tij of Rm, where (i,j) E G, with similarity ratios Uj such that (a) for each i, 1 < i < n, there is some j such that (i,j) E G, (b) for each i, {T{j(Jj)](i,j) E G} is a nonoverlapping family and (i) Ji2\J{T<AJ3WJ)eG} and (c) if the path component of G rooted at the vertex iy is a cycle: [iy,...,iq,iq+y = iy], then 9 (2) n 'w*+i < l k=l We will generalize the setting to ratio graph directed constructions later in the paper. Each geometric construction naturally determines a compact subset K of Rm. This set, which we will term the construction object, is pieced together by following the graph 67 and applying the maps coded by the edges to the corresponding sets. To make explicit how this is done we will first formulate how the pieces are generated. For each t, let 3£(Ji) be the space of compact subsets of Ji provided with the Hausdorff metric, pnReceived by the editors August 17, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 28C10.