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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2410.01663 |
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| _version_ | 1866929524385513472 |
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| author | Qiu, Jiahao Ye, Xiangdong |
| author_facet | Qiu, Jiahao Ye, Xiangdong |
| contents | For an abelian group $G$,
$\vec{g}=(g_1,\ldots,g_d)\in G^d$ and $ε=(ε(1),\ldots,ε(d))\in \{0,1\}^d$, let $\vec{g}\cdot ε=\prod_{i=1}^{d}g_i^{ε(i)}$. In this paper, it is shown that for a minimal system $(X,G)$ with $G$ being abelian, $(x,y)\in \mathbf{RP}^{[d]}$ if and only if there exists a sequence $\{\vec{g}_n\}_{n\in \mathbb{N}}\subseteq G^d$ and points $z_ε\in X,ε\in \{0,1\}^d$ with $z_{\vec{0}}=y$ such that for every $ε\in \{0,1\}^d\backslash\{ \vec{0}\}$, \[ \lim_{n\to\infty}(\vec{g}_n\cdotε)x= z_ε\quad \mathrm{and} \quad \lim_{n\to\infty}(\vec{g}_n\cdotε)^{-1}z_{\vec{1}}=z_{\vec{1}-ε}, \] where $\mathbf{RP}^{[d]}$ is the regionally proximal relation of order $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01663 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Veech's theorem of higher order Qiu, Jiahao Ye, Xiangdong Dynamical Systems For an abelian group $G$, $\vec{g}=(g_1,\ldots,g_d)\in G^d$ and $ε=(ε(1),\ldots,ε(d))\in \{0,1\}^d$, let $\vec{g}\cdot ε=\prod_{i=1}^{d}g_i^{ε(i)}$. In this paper, it is shown that for a minimal system $(X,G)$ with $G$ being abelian, $(x,y)\in \mathbf{RP}^{[d]}$ if and only if there exists a sequence $\{\vec{g}_n\}_{n\in \mathbb{N}}\subseteq G^d$ and points $z_ε\in X,ε\in \{0,1\}^d$ with $z_{\vec{0}}=y$ such that for every $ε\in \{0,1\}^d\backslash\{ \vec{0}\}$, \[ \lim_{n\to\infty}(\vec{g}_n\cdotε)x= z_ε\quad \mathrm{and} \quad \lim_{n\to\infty}(\vec{g}_n\cdotε)^{-1}z_{\vec{1}}=z_{\vec{1}-ε}, \] where $\mathbf{RP}^{[d]}$ is the regionally proximal relation of order $d$. |
| title | Veech's theorem of higher order |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2410.01663 |