×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] [Arn-Ber-Ito] P. Arnoux, V. Berthé and Sh. Ito, Discrete planes {ie206-1}, Jacobi-Perron algorithm and substitutions, preprint.
[2] [Arn-Ito] P. Arnoux and Sh. Ito,Pisot substitutions and Rauzy fractals, Bull. Soc. Math. Belg. (2001), to appear. · Zbl 1007.37001
[3] [Dek1] F. M. Dekking,Recurrent sets, Adv. Math.44 (1982), 78–104. · Zbl 0495.51017
[4] [Dek2] F. M. Dekking,Replicating superfigures and endomorphisms of free groups, J. Combin. Theory Ser. A32 (1982), 315–320. · Zbl 0492.05019
[5] [Ei-Ito] H. Ei and Sh. Ito,Decomposition theorem on invertible substitutions, Osaka J. Math.35 (1998), 821–834. · Zbl 0924.20040
[6] [Ito-Ohtsuki] Sh. Ito and M. Ohtsuki,Modified Jacoby-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms, Tokyo J. Math.16 (1993), 441–472. · Zbl 0805.11056
[7] [Ito-Kimura] Sh. Ito and M. Kimura,On Rauzy fractals, Japan J. Indust. Appl. Math.8 (1991), 461–486. · Zbl 0734.28010
[8] [Ito-Miz] Sh. Ito and M. Mizutani,Potato exchange transformations with fractal domains, preprint.
[9] [MaKaSo] W. Magnus, A. Karrass and D. Solitar,Combinatorial Group Theory, Wiley Interscience, New York, 1966.
[10] [Messaoudi] A. Messaoudi,Autour du fractal de Rauzy, Thèse, Université d’Aix-Marseille2 (1996).
[11] [Mig-See] F. Mignosi and P. Séébold,Morphismes sturmiens et règles de Rauzy, J. Théor. Nombres Bordeaux5 (1993), 221–233. · Zbl 0797.11029
[12] [Rauzy] G. Rauzy,Nombres algébriques et substitutions, Bull. Soc. Math. France110 (1982), 147–178. · Zbl 0522.10032
[13] [Wen-Wen] Z.-X. Wen and Z.-Y. Wen,Local isomorphisms of invertible substitutions, C. R. Acad. Sci. Paris, Série I318 (1994), 299–304. · Zbl 0812.11018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.