For any primitive substitution whose Perron eigenvalue is a Pisot unit, we construct a domain exchange that is measurably conjugate to the subshift. Additionally, we give a condition for the subshift to be a finite extension of a torus translation. For the particular case of weakly irreducible Pisot substitutions, we show that the subshift is either a finite extension of a torus translation or its eigenvalues are roots of unity. Furthermore, we provide an algorithm to compute eigenvalues of the subshift associated with any primitive pseudo-unimodular substitution.

中文翻译：

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原始替换的子移位的几何表示

对于任何 Perron 特征值是 Pisot 单位的原语替换，我们构造一个与子移位可测量地共轭的域交换。此外，我们给出了子平移是环面平移的有限扩展的条件。对于弱不可约皮索替换的特殊情况，我们证明子平移要么是环面平移的有限扩展，要么它的特征值是单位根。此外，我们提供了一种算法来计算与任何原始伪单模替换相关的子移位的特征值。