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Zbl 0931.11004
Holton, Charles; Zamboni, Luca Q.
Geometric realizations of substitutions. (English)
[J] Bull. Soc. Math. Fr. 126, No.2, 149-179 (1998). ISSN 0037-9484

A substitution rule on a finite alphabet induces a map on the space of the associated semi-infinite sequences of symbols. If the substitution is primitive, some iterate of this map has a fixed point, and the one-sided shift on the latter has simple ergodic properties. Under the additional assumption that the incidence matrix of the substitution has one eigenvalue inside the unit circle, the authors construct a representation of sequences in the complex plane, with the property that the substitution becomes a contraction, and the shift a finitely generated walk. The resulting limit sets are typically fractal. \par The constructs are simple and compelling, while the clear exposition and generous supply of examples will help the non-specialist to gain an overview of an interesting interdisciplinary subject. However, the elitist first sentence `\dots{} a free geometric exotic $\Bbb{F}_3$ action of an $\Bbb{R}$-tree\dots{}' (these terms are not used again in the paper), may have the unfortunate effect of alienating a large number of readers.
[F.Vivaldi (London)]

MSC 2000:: *11B85 Automata sequences
37B10 Symbolic dynamics
20E08 Groups acting on trees

Keywords: substitutions; limit sets; graph directed constructions

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