On the independence of shifts defined on $\mathbb{N}^d$ and trees
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915043947315200 |
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| author | Ban, Jung-Chao Lai, Guan-Yu |
| author_facet | Ban, Jung-Chao Lai, Guan-Yu |
| contents | In this paper, we study the independence of shifts defined on $\mathbb{N}^d$ ($\mathbb{N}^d$ shift) and trees (tree-shift). Firstly, for the completeness of the article, we provide a proof that an $\mathbb{N}^d$ shift has positive (topological) entropy if and only if it has an independence set with positive upper density. Secondly, we obtain that when the base shift $X$ is a hereditary shift, then the associated tree-shift $\mathcal{T}_X$ on an unexpandable tree has positive entropy if and only if it has an independence set with positive density. However, the independence of the tree-shift on an expandable tree differs from that of $\mathbb{N}^d$ shifts or tree-shifts on unexpandable trees. The boundary independence property is introduced and we prove that it is equivalent to the positive entropy of a tree-shift on an expandable tree. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_01049 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the independence of shifts defined on $\mathbb{N}^d$ and trees Ban, Jung-Chao Lai, Guan-Yu Dynamical Systems Combinatorics In this paper, we study the independence of shifts defined on $\mathbb{N}^d$ ($\mathbb{N}^d$ shift) and trees (tree-shift). Firstly, for the completeness of the article, we provide a proof that an $\mathbb{N}^d$ shift has positive (topological) entropy if and only if it has an independence set with positive upper density. Secondly, we obtain that when the base shift $X$ is a hereditary shift, then the associated tree-shift $\mathcal{T}_X$ on an unexpandable tree has positive entropy if and only if it has an independence set with positive density. However, the independence of the tree-shift on an expandable tree differs from that of $\mathbb{N}^d$ shifts or tree-shifts on unexpandable trees. The boundary independence property is introduced and we prove that it is equivalent to the positive entropy of a tree-shift on an expandable tree. |
| title | On the independence of shifts defined on $\mathbb{N}^d$ and trees |
| topic | Dynamical Systems Combinatorics |
| url | https://arxiv.org/abs/2412.01049 |