Fundamental group of tiles associated to quadratic canonical number systems

Rok, strany: 2008, 241 - 251

If $\mathbf{A}$ is a $2× 2$ expanding matrix with integral coefficients, and $\mathcal{D} \subset \mathbb{Z}²$ a complete set of coset representatives of $\mathbb{Z}²/\mathbf{A}\mathbb{Z}²$ with $|\det(\mathbf{A})|$ elements, then the set $\mathcal{T}$ defined by $\mathbf{A}\mathcal{T}=\mathcal{T}+\mathcal{D}$ is a self-affine plane tile of $\mathbb{R}²$, provided that its two-dimensional Lebesgue measure is positive. It was shown by Luo and Thuswaldner that the fundamental group of such a tile is either trivial or uncountable. To a quadratic polynomial $x²+Ax+B$, $A,B \in \mathbb{Z}$ such that $B≥2$ and $-1≤ A≤ B$, one can attach a tile $\mathcal{T}$. Akiyama and Thuswaldner proved the triviality of the fundamental group of this tile for $2A

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ISO 690:

Loridant, B. 2008. Fundamental group of tiles associated to quadratic canonical number systems. In Mathematica Slovaca, vol. 58, no.2, pp. 241-251. 0139-9918.

APA:

Loridant, B. (2008). Fundamental group of tiles associated to quadratic canonical number systems. Mathematica Slovaca, 58(2), 241-251. 0139-9918.