通过S-adic表示研究极小移位的连续特征值
Continuous eigenvalues of minimal subshifts via S-adic representations and coboundaries
AI总结：
本文通过S-adic表示研究极小移位的连续特征值，结合组合学与线性代数方法，提出特征化条件及非平凡上界不存在的充分条件，并恢复Tijdeman关于转移移位的最小因子复杂度结果。
摘要：
我们提供了对由满足自然温和条件的S-adic结构描述的极小符号动力系统连续特征值的特征化。在额外假设下，如有限字母秩或指令序列的决定性，这些特征化通过相关的局部上界序列表达。我们强调通过上界与扩展图之间的相互作用，组合学在研究连续特征值中的作用，并给出了若干非平凡字母-上界不存在的充分条件。进一步的结果包括在有界偏差背景下应用上界，特别是我们获得了原始替换移位的字母平衡的简单特征化。此外，我们恢复了Tijdeman关于具有有理数无关字母频率的极小转移移位的最小因子复杂度的结果。最后，我们利用线性代数对偶性，对特征值可能值的已知描述进行了细化，以基于基的度量进行表达。
英文摘要：
We provide characterizations of continuous eigenvalues for minimal symbolic dynamical systems. These characterizations rely on a description of the system in terms of $S$-adic structures (i.e. infinite compositions of morphisms) satisfying natural mild conditions, such as recognizability and primitivity. Under the additional assumptions of finite alphabet rank or decisiveness of the directive sequence, these characterizations involve sequences of letter coboundaries. We emphasize the role of combinatorics in the study of continuous eigenvalues through the interplay between letter coboundaries and extension graphs, and we provide several sets of sufficient conditions ensuring the triviality of letter coboundaries. These results are applied, among other settings, to linear involutions and to the Thue--Morse system in the rational base $3/2$. We also illustrate the versatility of the notion of letter coboundaries in the context of bounded symbolic discrepancy. In particular, we recover a simple characterization of letter balance for primitive substitutive subshifts. Finally, we refine known descriptions of the possible continuous eigenvalues in terms of the measures of the bases of the towers provided by the $S$-adic representation.
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