Shifts of Finite Type Obtained by Forbidding a Single Pattern
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| Format: | Preprint |
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2024
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| _version_ | 1866911244772966400 |
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| author | Chandgotia, Nishant Marcus, Brian Richey, Jacob Wu, Chengyu |
| author_facet | Chandgotia, Nishant Marcus, Brian Richey, Jacob Wu, Chengyu |
| contents | Given a finite word $w$, Guibas and Odlyzko (J. Combin. Theory Ser. A, 30, 1981, 183-208) showed that the autocorrelation polynomial $ϕ_w(t)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_09024 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shifts of Finite Type Obtained by Forbidding a Single Pattern Chandgotia, Nishant Marcus, Brian Richey, Jacob Wu, Chengyu Dynamical Systems Combinatorics Probability 2020 Primary 37B10, Secondary 60J10 Given a finite word $w$, Guibas and Odlyzko (J. Combin. Theory Ser. A, 30, 1981, 183-208) showed that the autocorrelation polynomial $ϕ_w(t)$ 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. |
| title | Shifts of Finite Type Obtained by Forbidding a Single Pattern |
| topic | Dynamical Systems Combinatorics Probability 2020 Primary 37B10, Secondary 60J10 |
| url | https://arxiv.org/abs/2409.09024 |