Invertible substitutions and local isomorphisms
[Substitutions inversibles et isomorphismes locaux]
Wen, Zhi-Xiong1 ; Wen, Zhi-Ying2 ; Wu, Jun1
1 Nonlinear Science Center, Department of Mathematics, Wuhan University, Wuhan 430072, China
2 Department of Mathematics, Tsinghua University, Beijing 10084, China
Comptes Rendus. Mathématique, Tome 334 (2002) no. 8, pp. 629-634.

Soient ϕ1 et ϕ2 deux substitutions primitives inversibles sur un alphabet de deux lettres. Soit ξϕ1 (resp. ξϕ2) un point fixe de ϕ1 (resp. ϕ2). Nous montrons que ξϕ1 et ξϕ2 sont localement isomorphes si et seulement s'il existe une substitution primitive inversible ϕ0 et deux entiers positifs m et n tels que Mϕ1=Mϕ0m et Mϕ2=Mϕ0n, où Mϕ est la matrice de la substitution ϕ.

Let ϕ1 and ϕ2 be two primitive invertible substitutions over a two-letter alphabet. Let ξϕ1 and ξϕ2 be fixed points of ϕ1 and ϕ2, respectively. We show that ξϕ1 and ξϕ2 are locally isomorphic if and only if there exists a primitive invertible substitution ϕ0 and two positive integers m and n such that Mϕ1=Mϕ0m and Mϕ2=Mϕ0n, where Mϕ is the substitutive matrix of the substitution ϕ.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02234-3
Wen, Zhi-Xiong 1 ; Wen, Zhi-Ying 2 ; Wu, Jun 1

1 Nonlinear Science Center, Department of Mathematics, Wuhan University, Wuhan 430072, China
2 Department of Mathematics, Tsinghua University, Beijing 10084, China 
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Wen, Zhi-Xiong; Wen, Zhi-Ying; Wu, Jun. Invertible substitutions and local isomorphisms. Comptes Rendus. Mathématique, Tome 334 (2002) no. 8, pp. 629-634. doi : 10.1016/S1631-073X(02)02234-3. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02234-3/

[1] Berstel, J. Recent results in Sturmian words, Developments in Language Theory, II (Magdeburg, 1995), World Sci. Publishing, River Edge, NJ, 1996, pp. 13-24

[2] Cobham, A. On the base-dependence of sets of numbers recognizable by finite automata, Math. Systems Theory, Volume 3 (1969), pp. 186-192

[3] Durand, F. A generalization of Cobham's theory, Theory Comput. Systems, Volume 31 (1998), pp. 169-185

[4] Fagnot, I. On the subword equivalence problem for morphic words, Discrete Appl. Math., Volume 75 (1997), pp. 231-253

[5] Fagnot, I. Sur les facteurs des mots automatiques, Theoret. Comput. Sci., Volume 172 (1997), pp. 67-89

[6] Lothaire, M. Combinatorics on words, Encyclopedia of Mathematics and its Applications, 17, Addison-Wesley, 1983

[7] Luck, J.M.; Godrèche, C.; Janner, A.; Janssen, T. The nature of the atomic surfaces of quasiperiodic self-similar structures, J. Phys. A, Volume 26 (1993) no. 8, pp. 1951-1999

[8] Mignosi, F.; Séébold, P. Morphismes Sturmians et règles de Rauzy, J. de Théorie des Nombres de Bordeaux, Volume 5 (1993), pp. 221-233

[9] Queffélec, M. Substitution Dynamical Systems – Spectral Analysis, Lecture Notes in Math., 1294, Springer-Verlag, 1987

[10] Wen, Z.X.; Wen, Z.Y. Local isomorphisms of the invertible substitutions, C. R. Acad. Sci. Paris, Série I, Volume 318 (1994), pp. 299-304

[11] Wen, Z.X.; Wen, Z.Y. Some properties of the singular words of the Fibonacci word, European J. Combin., Volume 15 (1994), pp. 587-598

[12] Wen, Z.X.; Wen, Z.Y. Factor properties of the infinite words generated by a class of invertible substitutions, 5th Conference Formal Power Series and Algebraic Combinatorics, Florence, 1993, pp. 455-466

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☆ Supported by the Special Funds for Major State Basic Research Projects of China and Morningside Center of Mathematics (CAS).