Tilings associated with non-Pisot matrices
Annales de l'Institut Fourier, Volume 56 (2006) no. 7, p. 2391-2435
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.
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.
DOI : https://doi.org/10.5802/aif.2244
Classification:  37B50,  52C20,  11R06,  15A15
Keywords: Tilings, substitutions, non-Pisot property, Binet-Cauchy theorem
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {7},
     year = {2006},
     pages = {2391-2435},
     doi = {10.5802/aif.2244},
     zbl = {1142.15015},
     mrnumber = {2290785},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2006__56_7_2391_0}
}
Tilings associated with non-Pisot matrices. Annales de l'Institut Fourier, Volume 56 (2006) no. 7, pp. 2391-2435. doi : 10.5802/aif.2244. https://aif.centre-mersenne.org/item/AIF_2006__56_7_2391_0/

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