Tilings associated with non-Pisot matrices

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

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

Supposons que $A\in G{l}_d\left(\mathbb{Z}\right)$ ait un sous-espace d’extension bidimensionnel $E^u$, satisfaisant une condition de régularité, appelée “bonne étoile”, et telle que $A^{*}\ge 0$, où $A^{*}$ est un composé orienté. Un morphisme $\theta$ du groupe libre sur $\{1,2,\cdots ,d\}$ est une non-abélianisation de $A$ si sa matrice de structure est $A$. Nous prouvons qu’il existe une substitution de pavage $\Theta$ dont la substitution de frontière $\theta =\partial \Theta$ est une non-abélianisation de $A$. Une telle substitution de pavage $\theta$ donne un pavage “auto-affine” de $E^u\sim {ℝ}^2$ avec pour expansion $A_u{:=A|}_{E_u}\in G{L}_2\left(ℝ\right)$. Dans la dernière section nous trouvons des conditions sur $A$ de sorte que $A^{*}$ n’ait pas de coefficients négatifs.

Suppose $A\in G{l}_d\left(\mathbb{Z}\right)$ has a 2-dimensional expanding subspace $E^u$, satisfies a regularity condition, called “good star”, and has $A^{*}\ge 0$, where $A^{*}$ is an oriented compound of $A$. A morphism $\theta$ of the free group on $\{1,2,\cdots ,d\}$ is called a non-abelianization of $A$ if it has structure matrix $A$. We show that there is a tiling substitution $\Theta$ whose “boundary substitution” $\theta =\partial \Theta$ is a non-abelianization of $A$. Such a tiling substitution $\Theta$ leads to a self-affine tiling of $E^u\sim {ℝ}^2$ with $A_u{:=A|}_{E_u}\in G{L}_2\left(ℝ\right)$ as its expansion. In the last section we find conditions on $A$ so that $A^{*}$ has no negative entries.

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

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