论文标题
关于SOFIC偏移的轨道生长的简短说明
A short note on the orbit growth of sofic shifts
论文作者
论文摘要
Sofic Shift是一个偏移空间，该移位空间由标记图的路径的Bi-Infinite标签组成。作为一个动态系统，其闭合轨道的分布可能表明空间的复杂性。为此，引入了主要的轨道和梅尔滕斯的轨道计数功能，以描述封闭轨道的生长。这些计数函数的渐近行为可以暗示该空间的Artin-Mazur Zeta功能的分类。尽管具有封闭形式的表达，但Zeta函数还是通过几个签名的子集矩阵隐式表示，这使得其分析性的研究似乎很困难。在本文中，我们将证明通过其ZETA函数进行SOFIC转移的计数函数的渐近行为。这涉及研究上述矩阵的特性。令人惊讶的是，证据相当短，只使用有关索福转变的众所周知的事实，尤其是在其最小的右分辨率演示中。
A sofic shift is a shift space consisting of bi-infinite labels of paths from a labelled graph. Being a dynamical system, the distribution of its closed orbits may indicate the complexity of the space. For this purpose, prime orbit and Mertens' orbit counting functions are introduced as a way to describe the growth of the closed orbits. The asymptotic behaviours of these counting functions can be implied from the analiticity of the Artin-Mazur zeta function of the space. Despite having a closed-form expression, the zeta function is expressed implicitly in terms of several signed subset matrices, and this makes the study on its analyticity to be seemingly difficult. In this paper, we will prove the asymptotic behaviours of the counting functions for a sofic shift via its zeta function. This involves investigating the properties of the said matrices. Suprisingly, the proof is rather short and only uses well-known facts about a sofic shift, especially on its minimal right-resolving presentation.
