## 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

### Keywords:

morphisms; Rauzy fractal; substitutions

### Citations:

Zbl 1007.37001; Zbl 0918.11048; Zbl 1017.11006
Full Text:

### References:

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