Mathematica Slovaca | |
Fundamental group of tiles associated to quadratic canonical number systems | |
Benoit Loridant1  | |
关键词: tiles; fundamental group; automata; | |
DOI : 10.2478/s12175-008-0070-7 | |
学科分类:数学(综合) | |
来源: Slovenska Akademia Vied * Matematicky Ustav / Slovak Academy of Sciences, Mathematical Institute | |
【 摘 要 】
If A is a 2 × 2 expanding matrix with integral coefficients, and ⊂ ℤ2 a complete set of coset representatives of ℤ2/Aℤ2 with |det(A)| elements, then the set â„ defined by Aâ„ = â„ + is a self-affine plane tile of â„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 x 2 + Ax + B, A, B ∈ ℤ such that B ≥ 2 and −1 ≤ A ≤ B, one can attach a tile â„. Akiyama and Thuswaldner proved the triviality of the fundamental group of this tile for 2A < B + 3, by showing that a tile of this class is homeomorphic to a closed disk. The case 2A ≥ B + 3 is treated here by using the criterion given by Luo and Thuswaldner.
【 授权许可】
Unknown
【 预 览 】
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