Tilings associated with non-Pisot matrices
[Pavages associés à des matrices non-Pisot]
Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2391-2435.

Supposons que AGl d () ait un sous-espace d’extension bidimensionnel E u , satisfaisant une condition de régularité, appelée “bonne étoile”, et telle que A * 0, où A * est un composé orienté. Un morphisme θ du groupe libre sur {1,2,,d} est une non-abélianisation de A si sa matrice de structure est A. Nous prouvons qu’il existe une substitution de pavage Θ dont la substitution de frontière θ=Θ est une non-abélianisation de A. Une telle substitution de pavage θ donne un pavage “auto-affine” de E u 2 avec pour expansion A u :=A| E u GL 2 (). Dans la dernière section nous trouvons des conditions sur A de sorte que A * n’ait pas de coefficients négatifs.

Suppose AGl d () has a 2-dimensional expanding subspace E u , satisfies a regularity condition, called “good star”, and has A * 0, where A * is an oriented compound of A. A morphism θ of the free group on {1,2,,d} is called a non-abelianization of A if it has structure matrix A. We show that there is a tiling substitution Θ whose “boundary substitution” θ=Θ is a non-abelianization of A. Such a tiling substitution Θ leads to a self-affine tiling of E u 2 with A u :=A| E u GL 2 () as its expansion. In the last section we find conditions on A so that A * has no negative entries.

DOI : 10.5802/aif.2244
Classification : 37B50, 52C20, 11R06, 15A15
Keywords: Tilings, substitutions, non-Pisot property, Binet-Cauchy theorem
Mot clés : pavages, substitutions, properté non-Pisot, théorème de Binet-Cauchy
Furukado, Maki 1 ; Ito, Shunji 2 ; Robinson, E. Arthur Jr 3

1 Yokohama National University Faculty of Business Administration 79-4, Tokiwadai, Hodogaya-Ku Yokohama 240-8501 (Japan)
2 Kanazawa University Graduate School of Natural Science & Technology Kakuma-machi Kanazawa 920-1192 (Japan)
3 George Washington University Department of Mathematics Washington, DC 20052 (USA)
@article{AIF_2006__56_7_2391_0,
     author = {Furukado, Maki and Ito, Shunji and Robinson, E. Arthur Jr},
     title = {Tilings associated with {non-Pisot} matrices},
     journal = {Annales de l'Institut Fourier},
     pages = {2391--2435},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {7},
     year = {2006},
     doi = {10.5802/aif.2244},
     mrnumber = {2290785},
     zbl = {1142.15015},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2244/}
}
TY  - JOUR
AU  - Furukado, Maki
AU  - Ito, Shunji
AU  - Robinson, E. Arthur Jr
TI  - Tilings associated with non-Pisot matrices
JO  - Annales de l'Institut Fourier
PY  - 2006
SP  - 2391
EP  - 2435
VL  - 56
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2244/
DO  - 10.5802/aif.2244
LA  - en
ID  - AIF_2006__56_7_2391_0
ER  - 
%0 Journal Article
%A Furukado, Maki
%A Ito, Shunji
%A Robinson, E. Arthur Jr
%T Tilings associated with non-Pisot matrices
%J Annales de l'Institut Fourier
%D 2006
%P 2391-2435
%V 56
%N 7
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2244/
%R 10.5802/aif.2244
%G en
%F AIF_2006__56_7_2391_0
Furukado, Maki; Ito, Shunji; Robinson, E. Arthur Jr. Tilings associated with non-Pisot matrices. Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2391-2435. doi : 10.5802/aif.2244. https://aif.centre-mersenne.org/articles/10.5802/aif.2244/

[1] Ahlfors, L. V. Complex Analysis, McGraw-Hill, 1978 | MR | Zbl

[2] Aitken, A. C. Determinants and Matrices, Oliver and Boyd, Ltd., 1956 | Zbl

[3] Arnoux, P.; Berthé, V.; Ei, H.; Ito, S. Tilings, quasicrystals, discrete planes, generalized substitutions, and multidimensional continued fractions, Discrete models: combinatorics, computation, and geometry (Discrete Math. Theor. Comput. Sci. Proc., AA), Maison Inform. Math. Discrèt., Paris, 2001 (059-078) | MR | Zbl

[4] Arnoux, P.; Ito, S. Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. (2001), pp. 181-207 | MR | Zbl

[5] de Bruijn, N. G. Algebraic theory of Penrose’s nonperiodic tilings of the plane. I, II, Nederl. Akad. Wetensch. Indag. Math., Volume 43 (1981) no. 1, p. 39-52, 53-66 | Zbl

[6] Ei, H. Some properties of invertible substitutions of rank d, and higher dimensional substitutions, Osaka J. Math., Volume 40 (2003) no. 2, pp. 543-562 | MR | Zbl

[7] Ei, H.; Ito, S. Tilings from some non-irreducible, Pisot substitutions, Discrete Math. Theor. Comput. Sci., Volume 7 (2005) no. 1, pp. 81-121 | MR | Zbl

[8] Frank, N. P.; Robinson, E. A. Jr. Generalized β -expansions, substitution tilings and local finiteness (to appear in Transactions Amer. Math. Soc.) | Zbl

[9] Furukado, M. Tiling from non-Pisot unimodular matrices (to appear in Hirosihima Math. J.) | MR | Zbl

[10] Furukado, M.; Ito, S. Connected Markov Partitions of group automorphisms ands Rauzy fractals Substitution and its applicatoin : Research Report Grant-in-Aid scientific Research (c)(2), (project number 09640291, Japan (2002), p. 41-92

[11] Harriss, E. O.; Lamb, J. S. W. Canonical substitutions tilings of Ammann-Beenker type, Theoret. Comput. Sci., Volume 319 (2004) no. 1-3, pp. 241-279 | DOI | MR | Zbl

[12] Ito, S.; Ohtsuki, M. Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms, Tokyo J. Math., Volume 16 (1993) no. 2, pp. 441-472 | DOI | MR | Zbl

[13] Kenyon, R. Self-similar tilings, Princeton University (1990) (Ph. D. Thesis)

[14] Lind, D.; Marcus, B. An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[15] Rauzy, G. Nombres algébriques et substitutions, Bull. Soc. Math. France, Volume 110 (1982) no. 2, pp. 147-178 | Numdam | MR | Zbl

[16] Robinson, E. A. Jr.; Williams, Susan G. Symbolic dynamics and tilings of d , Symbolic dynamics and its applications (Proc. Sympos. Appl. Math.), Volume 60, Amer. Math. Soc., Providence, RI, 2004, pp. 81-119 | MR | Zbl

[17] Senechal, M. Quasicrystals and geometry, Cambridge University Press, Cambridge, 1995 | MR | Zbl

Cité par Sources :