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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2501.08438 |
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| _version_ | 1866909562896908288 |
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| author | Ramirez, Cristian Somers, Amy |
| author_facet | Ramirez, Cristian Somers, Amy |
| contents | The $S$-gap shifts have a dynamically and combinatorially rich structure. Dynamical properties of the $S$-gap shift can be related to the properties of the set $S$. This interplay is particularly interesting when $S$ is not syndetic such as when $S$ is the set of prime numbers or when $S=\{2^n\}$. It is a well known result that the entropy of the $S$-gap shift is given by $h(X) = \log λ$, where $λ>0$ is the unique solution to the equation $\sum_{n \in S} λ^{-(n+1)}=1$. Fix a point $w$ of the full shift $\{1,2, \dots, k\}^\mathbb{Z}$. We introduce the $(S,w)$-gap shift which is a generalization of the $S$-gap shift consisting of sequences in $\{0,1, \dots, k\}^\mathbb{Z}$ in which any two $0$'s are separated by a word $u$ appearing in $w$ such that $|u|\in S$. We extend the formula for the entropy of the $S$-gap shift to a formula describing the entropy of this new class of shift spaces. Additionally we investigate the dynamical properties including irreducibility and mixing of this generalization of the $S$-gap shift. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08438 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | (S,w)-Gap Shifts and Their Entropy Ramirez, Cristian Somers, Amy Dynamical Systems 37B10, 37A35 The $S$-gap shifts have a dynamically and combinatorially rich structure. Dynamical properties of the $S$-gap shift can be related to the properties of the set $S$. This interplay is particularly interesting when $S$ is not syndetic such as when $S$ is the set of prime numbers or when $S=\{2^n\}$. It is a well known result that the entropy of the $S$-gap shift is given by $h(X) = \log λ$, where $λ>0$ is the unique solution to the equation $\sum_{n \in S} λ^{-(n+1)}=1$. Fix a point $w$ of the full shift $\{1,2, \dots, k\}^\mathbb{Z}$. We introduce the $(S,w)$-gap shift which is a generalization of the $S$-gap shift consisting of sequences in $\{0,1, \dots, k\}^\mathbb{Z}$ in which any two $0$'s are separated by a word $u$ appearing in $w$ such that $|u|\in S$. We extend the formula for the entropy of the $S$-gap shift to a formula describing the entropy of this new class of shift spaces. Additionally we investigate the dynamical properties including irreducibility and mixing of this generalization of the $S$-gap shift. |
| title | (S,w)-Gap Shifts and Their Entropy |
| topic | Dynamical Systems 37B10, 37A35 |
| url | https://arxiv.org/abs/2501.08438 |