Tilings associated with non-Pisot matrices

Furukado, Maki; Ito, Shunji; Robinson, E. Arthur Jr

doi:10.5802/aif.2244

Tilings associated with non-Pisot matrices
[Pavages associés à des matrices non-Pisot]

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)

Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2391-2435.

Résumé
Abstract

Supposons que $A \in G l_{d} (ℤ)$ ait un sous-espace d’extension bidimensionnel $E^{u}$ , satisfaisant une condition de régularité, appelée “bonne étoile”, et telle que $A^{*} \geq 0$ , où $A^{*}$ est un composé orienté. Un morphisme $θ$ du groupe libre sur ${1, 2, \dots, d}$ 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 $θ = \partial Θ$ est une non-abélianisation de $A$ . Une telle substitution de pavage $θ$ donne un pavage “auto-affine” de $E^{u} \sim ℝ^{2}$ avec pour expansion $A_{u} {: = A |}_{E_{u}} \in G L_{2} (ℝ)$ . Dans la dernière section nous trouvons des conditions sur $A$ de sorte que $A^{*}$ n’ait pas de coefficients négatifs.

Suppose $A \in G l_{d} (ℤ)$ has a 2-dimensional expanding subspace $E^{u}$ , satisfies a regularity condition, called “good star”, and has $A^{*} \geq 0$ , where $A^{*}$ is an oriented compound of $A$ . A morphism $θ$ of the free group on ${1, 2, \dots, d}$ 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” $θ = \partial Θ$ is a non-abelianization of $A$ . Such a tiling substitution $Θ$ leads to a self-affine tiling of $E^{u} \sim ℝ^{2}$ with $A_{u} {: = A |}_{E_{u}} \in G L_{2} (ℝ)$ as its expansion. In the last section we find conditions on $A$ so that $A^{*}$ has no negative entries.

MR Zbl

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

Affiliations des auteurs :

Furukado, Maki ¹ ; Ito, Shunji ² ; Robinson, E. Arthur Jr ³

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     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},
     zbl = {1142.15015},
     mrnumber = {2290785},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.2244/}
}

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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://www.numdam.org/articles/10.5802/aif.2244/

Bibliographie
Cité par

[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 $ℝ^{d}$ , 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 :