Shifts of Finite Type Obtained by Forbidding a Single Pattern

Nishant Chandgotia, Brian Marcus, Jacob Richey, Chengyu Wu
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Abstract

Given a finite word $w$, Guibas and Odlyzko (J. Combin. Theory Ser. A, 30, 1981, 183-208) showed that the autocorrelation polynomial $\phi_w(t)$ of $w$, which records the set of self-overlaps of $w$, explicitly determines for each $n$, the number $|B_n(w)|$ of words of length $n$ that avoid $w$. We consider this and related problems from the viewpoint of symbolic dynamics, focusing on the setting of $X_{\{w\}}$, the space of all bi-infinite sequences that avoid $w$. We first summarize and elaborate upon (J. Combin. Theory Ser. A, 30, 1981, 183-208) and other work to show that the sequence $|B_n(w)|$ is equivalent to several invariants of $X_{\{w\}}$. We then give a finite-state labeled graphical representation $L_w$ of $X_{\{w\}}$ and show that $w$ can be recovered from the graph isomorphism class of the unlabeled version of $L_w$. Using $L_w$, we apply ideas from probability and Perron-Frobenius theory to obtain results comparing features of $X_{\{w\}}$ for different $w$. Next, we give partial results on the problem of classifying the spaces $X_{\{w\}}$ up to conjugacy. Finally, we extend some of our results to spaces of multi-dimensional arrays that avoid a given finite pattern.
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通过禁止单一模式获得的有限类型转变
给定一个有限词 $w$,Guibas 和 Odlyzko (J. Combin. Theory Ser. A, 30,1981, 183-208) 发现,记录了 $w$ 的自重叠集合的 $w$ 的自相关多项式 $\phi_w(t)$,明确地决定了每个 $n$ 的长度为 $n$ 的词中避开 $w$ 的词的数目 $|B_n(w)|$。我们从符号动力学的角度来考虑这个问题及相关问题,重点是 $X_{\{w\}}$,即所有避开 $w$ 的双无限序列的空间。我们首先总结并阐述了 (J. Combin. Theory Ser. A, 30, 1981,183-208)和其他工作,以证明序列 $|B_n(w)|$ 等价于 $X_{\{w\}}$ 的几个不变式。然后,我们给出了$X_{\{w\}}$的有限状态标注图表示$L_w$,并证明$w$可以从未标明版本的$L_w$的图同构类中得到。利用$L_w$,我们应用概率论和佩伦-弗罗贝尼斯理论的思想,得到了比较不同$w$下$X_{\{w\}}$特征的结果。接下来,我们给出了对直到共轭的空间 $X_{\{w\}$ 的分类问题的部分结果。最后,我们将部分结果扩展到避免给定有限模式的多维阵列空间。
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