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Geometric realizations of substitutions. (English) Zbl 0931.11004
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.
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 ‘… a free geometric exotic \(\mathbb{F}_3\) action of an \(\mathbb{R}\)-tree…’ (these terms are not used again in the paper), may have the unfortunate effect of alienating a large number of readers.
Reviewer: F.Vivaldi (London)

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