# Tilings associated with non-Pisot matrices

Maki Furukado^{[1]}; Shunji Ito^{[2]}; E. Arthur Robinson^{[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 (2006)

- Volume: 56, Issue: 7, page 2391-2435
- ISSN: 0373-0956

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topFurukado, Maki, Ito, Shunji, and Robinson, E. Arthur. "Tilings associated with non-Pisot matrices." Annales de l’institut Fourier 56.7 (2006): 2391-2435. <http://eudml.org/doc/10208>.

@article{Furukado2006,

abstract = {Suppose $A\in Gl_d(\mathbb\{Z\})$ 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 $\lbrace 1,2,\dots ,d\rbrace $ 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 $E^u\sim \mathbb\{R\}^2$ with $A_u:=A|_\{E_u\}\in GL_2(\mathbb\{R\})$ as its expansion. In the last section we find conditions on $A$ so that $A^*$ has no negative entries.},

affiliation = {Yokohama National University Faculty of Business Administration 79-4, Tokiwadai, Hodogaya-Ku Yokohama 240-8501 (Japan); Kanazawa University Graduate School of Natural Science & Technology Kakuma-machi Kanazawa 920-1192 (Japan); George Washington University Department of Mathematics Washington, DC 20052 (USA)},

author = {Furukado, Maki, Ito, Shunji, Robinson, E. Arthur},

journal = {Annales de l’institut Fourier},

keywords = {Tilings; substitutions; non-Pisot property; Binet-Cauchy theorem; tiling substitution; non-Pisot condition; deBruijn diagram; Ammann matrix; hyperbolic matrix; structure matrix},

language = {eng},

number = {7},

pages = {2391-2435},

publisher = {Association des Annales de l’institut Fourier},

title = {Tilings associated with non-Pisot matrices},

url = {http://eudml.org/doc/10208},

volume = {56},

year = {2006},

}

TY - JOUR

AU - Furukado, Maki

AU - Ito, Shunji

AU - Robinson, E. Arthur

TI - Tilings associated with non-Pisot matrices

JO - Annales de l’institut Fourier

PY - 2006

PB - Association des Annales de l’institut Fourier

VL - 56

IS - 7

SP - 2391

EP - 2435

AB - Suppose $A\in Gl_d(\mathbb{Z})$ 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 $\lbrace 1,2,\dots ,d\rbrace $ 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 $E^u\sim \mathbb{R}^2$ with $A_u:=A|_{E_u}\in GL_2(\mathbb{R})$ as its expansion. In the last section we find conditions on $A$ so that $A^*$ has no negative entries.

LA - eng

KW - Tilings; substitutions; non-Pisot property; Binet-Cauchy theorem; tiling substitution; non-Pisot condition; deBruijn diagram; Ammann matrix; hyperbolic matrix; structure matrix

UR - http://eudml.org/doc/10208

ER -

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