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)
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     title = {Tilings associated with {non-Pisot} matrices},
     journal = {Annales de l'Institut Fourier},
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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. http://www.numdam.org/articles/10.5802/aif.2244/

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