Recognizability for sequences of morphisms
Recognizability for sequences of morphisms
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DOI：
10.1017/etds.2017.144
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发表时间：
2019-11-01
影响因子：
0.9
通讯作者：
Yassawi, Reem
中科院分区：
文献类型：
--
作者：
Berthe, Valerie;Steiner, Wolfgang;Yassawi, Reem
我们研究了自由么半群态射σ：A* -> B* 的可识别性的不同概念。当B-Z中的每个（非周期性）点最多允许一个带有单词alpha（a）的平铺时，完全可识别性发生，a是A的元素。这比替换sigma的可识别性的经典概念更强：A -> A*，其中平铺必须与替换的语言兼容。我们表明，如果垂直酒吧A垂直酒吧= 2，或者如果西格玛的关联矩阵有秩垂直酒吧A垂直酒吧，或者如果西格玛是置换的，那么西格玛是完全可识别的。接下来，我们调查的经典概念的可识别性和改善早期的结果Mosse [Puissances德mots等reconnaissabilite des points fixes d 'une substitution.理论计算。Sci. 99（2）（1992），327-334]和Bezuglyi等人[Apperiodic substitution systems and their Bratteli diagrams.二神日& Dynam。Sys. 29（1）（2009），37-72]，通过表明任何替换对于其替换移位中的非周期点都是可识别的。最后，我们定义可识别性，并最终识别定义一个S-进移位的态射序列。我们证明了一个序列的有界大小的字母表上的态射，连续态射的成分是增长的所有字母，最终是可识别的非周期点。我们提供的例子最终可识别的，但不能识别，序列的态射，序列的态射是最终不能识别。作为一个应用程序，对于一个可识别的序列的态射，我们得到了几乎无处不在的双射对应的S-adic移位，它产生的，和可测量的Bratteli-Vershik动力系统，它定义。
We investigate different notions of recognizability for a free monoid morphism sigma : A* -> B*. Full recognizability occurs when each (aperiodic) point in B-Z admits at most one tiling with words alpha(a), a is an element of A. This is stronger than the classical notion of recognizability of a substitution sigma : A -> A*, where the tiling must be compatible with the language of the substitution. We show that if vertical bar A vertical bar = 2, or if sigma's incidence matrix has rank vertical bar A vertical bar, or if sigma is permutative, then sigma is fully recognizable. Next we investigate the classical notion of recognizability and improve earlier results of Mosse [Puissances de mots et reconnaissabilite des points fixes d'une substitution. Theoret. Comput. Sci. 99(2) (1992), 327-334] and Bezuglyi et al [Aperiodic substitution systems and their Bratteli diagrams. Ergod. Th. & Dynam. Sys. 29(1) (2009), 37-72], by showing that any substitution is recognizable for aperiodic points in its substitutive shift. Finally we define recognizability and also eventual recognizability for sequences of morphisms which define an S-adic shift. We prove that a sequence of morphisms on alphabets of bounded size, such that compositions of consecutive morphisms are growing on all letters, is eventually recognizable for aperiodic points. We provide examples of eventually recognizable, but not recognizable, sequences of morphisms, and sequences of morphisms which are not eventually recognizable. As an application, for a recognizable sequence of morphisms, we obtain an almost everywhere bijective correspondence between the S-adic shift it generates, and the measurable Bratteli-Vershik dynamical system that it defines.