Geometric theory of unimodular Pisot substitutions
Geometric theory of unimodular Pisot substitutions
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DOI：
10.1353/ajm.2006.0037
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发表时间：
2006-10
影响因子：
1.7
通讯作者：
M. Barge;J. Kwapisz
中科院分区：
文献类型：
--
作者：
M. Barge;J. Kwapisz
我们关注有限字母表上与替换φ相关联的平铺流T。我们的重点是在替换是么模皮索，即，它们的矩阵是幺模的，除了Perron特征值λ 1外，所有特征值都严格在单位圆内。其动机是由（仍然开放的）猜想提供的，该猜想断言对于任何这样的φ，T具有纯离散谱。给出了纯离散谱的一些充要条件，包括：正则环面映射的内射性（几何实现），几何重合条件，T与对偶Rd-1作用的（部分）对易，Rauzy分形的测度与平铺性质，以及具体算法.其中有些是原创的，有些已经在文献中出现过--仅仅作为充分条件--但它们都来自于一个统一的方法，该方法基于新的装置：φ的串空间F φ。必要性的证明取决于确定T的离散谱作为相关的Kronecker湍流的离散谱。
We are concerned with the tiling flow T associated to a substitution φ over a finite alphabet. Our focus is on substitutions that are unimodular Pisot, i.e., their matrix is unimodular and has all eigenvalues strictly inside the unit circle with the exception of the Perron eigenvalue λ 1. The motivation is provided by the (still open) conjecture asserting that T has pure discrete spectrum for any such φ. We develop a number of necessary and sufficient conditions for pure discrete spectrum, including: injectivity of the canonical torus map (the geometric realization), Geometric Coincidence Condition, (partial) commutation of T and the dual R d -1 -action, measure and tiling properties of Rauzy fractals, and concrete algorithms. Some of these are original and some have already appeared in the literature-as sufficient conditions only-but they all emerge from a unified approach based on the new device: the strand space F φ of φ. The proof of the necessity hinges on determination of the discrete spectrum of T as that of the associated Kronecker toral flow.