Tilings associated with non-Pisot matrices

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

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

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

Annales de l'Institut Fourier, Tome 56 (2006), p. 2391-2435 / Harvested from Numdam

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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.

Publié le : 2006-01-01

MR 2290785 | Zbl 1142.15015

DOI : https://doi.org/10.5802/aif.2244
Classification: 37B50, 52C20, 11R06, 15A15
Mots clés: pavages, substitutions, properté non-Pisot, théorème de Binet-Cauchy

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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},
     volume = {56},
     year = {2006},
     pages = {2391-2435},
     doi = {10.5802/aif.2244},
     zbl = {1142.15015},
     mrnumber = {2290785},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 2391-2435. doi : 10.5802/aif.2244. http://gdmltest.u-ga.fr/item/AIF_2006__56_7_2391_0/

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