Characterization of the tree cycles with minimum positive entropy for any period
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915484668002304 |
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| author | Juher, David Mañosas, Francesc Rojas, David |
| author_facet | Juher, David Mañosas, Francesc Rojas, David |
| contents | Consider, for any integer $n\ge3$, the set $\text{Pos}_n$ of all $n$-periodic tree patterns with positive topological entropy and the set $\text{Irr}_n\subset\text{Pos}_n$ of all $n$-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families $\text{Pos}_n$, $\text{Irr}_n$ and $\text{Pos}_n\setminus\text{Irr}_n$. Let $λ_n$ be the unique real root of the polynomial $x^n-2x-1$ in $(1,+\infty)$. We explicitly construct an irreducible $n$-periodic tree pattern $\mathcal{Q}_n$ whose entropy is $\log(λ_n)$. We prove that this entropy is minimum in $\text{Pos}_n$. Since the pattern $\mathcal{Q}_n$ is irreducible, $\mathcal{Q}_n$ also minimizes the entropy in the family $\text{Irr}_n$. We also prove that the minimum positive entropy in the set $\text{Pos}_n\setminus\text{Irr}_n$ (which is nonempty only for composite integers $n\ge6$) is $\log(λ_{n/p})/p$, where $p$ is the least prime factor of $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_14862 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Characterization of the tree cycles with minimum positive entropy for any period Juher, David Mañosas, Francesc Rojas, David Dynamical Systems 37E15, 37E25 Consider, for any integer $n\ge3$, the set $\text{Pos}_n$ of all $n$-periodic tree patterns with positive topological entropy and the set $\text{Irr}_n\subset\text{Pos}_n$ of all $n$-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families $\text{Pos}_n$, $\text{Irr}_n$ and $\text{Pos}_n\setminus\text{Irr}_n$. Let $λ_n$ be the unique real root of the polynomial $x^n-2x-1$ in $(1,+\infty)$. We explicitly construct an irreducible $n$-periodic tree pattern $\mathcal{Q}_n$ whose entropy is $\log(λ_n)$. We prove that this entropy is minimum in $\text{Pos}_n$. Since the pattern $\mathcal{Q}_n$ is irreducible, $\mathcal{Q}_n$ also minimizes the entropy in the family $\text{Irr}_n$. We also prove that the minimum positive entropy in the set $\text{Pos}_n\setminus\text{Irr}_n$ (which is nonempty only for composite integers $n\ge6$) is $\log(λ_{n/p})/p$, where $p$ is the least prime factor of $n$. |
| title | Characterization of the tree cycles with minimum positive entropy for any period |
| topic | Dynamical Systems 37E15, 37E25 |
| url | https://arxiv.org/abs/2310.14862 |