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Higher dimensional extensions of substitutions and their dual maps. (English) Zbl 0987.11013

The authors define \(k\)-dimensional extensions for substitutions (morphisms) on a \(d\)-letter alphabet. These are generalizations of 1-dimensional extensions (which act on the space of formal sums of weighted unit segments in \(\mathbb{Z}^d\)). They study in particular the case where the morphisms are unimodular, resp. hyperbolic. As the authors note in their conclusion, there probably is a homology theory underlying their constructions, but it has not been found yet.
Two remarks to end this review: the authors call the morphism \(1\to 121\), \(2\to 12\) the Fibonacci substitution. This morphism is actually the square of the morphism \(1\to 12\), \(2\to 1\), which is classically called the Fibonacci substitution.
References have appeared: P. Arnoux, V. Berthé and S. Ito, Discrete planes, \(\mathbb{Z}^2\)-actions, Jacobi-Perron algorithm and substitutions, Ann. Inst. Fourier 52, 305–349 (2002; Zbl 1017.11006) and P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. – Simon Stevin 8, 181–207 (2001; Zbl 1007.37001).
Finally note that Reference [Messaoudi] could be completed or replaced by his paper in J. Théor. Nombres Bordx. 10, No. 1, 135–162 (1998; Zbl 0918.11048)].

MSC:

11B85 Automata sequences
37B10 Symbolic dynamics
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28A80 Fractals
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[2] [Arn-Ito] P. Arnoux and Sh. Ito,Pisot substitutions and Rauzy fractals, Bull. Soc. Math. Belg. (2001), to appear. · Zbl 1007.37001
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