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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.11251 |
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| _version_ | 1866914802058657792 |
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| author | Ye, Xiangdong Yu, Jiaqi |
| author_facet | Ye, Xiangdong Yu, Jiaqi |
| contents | For a dynamical system $(X,T)$, $d\in\mathbb{N}$ and distinct non-constant integral polynomials $p_1,\ldots, p_d$ vanishing at $0$, the notion of regionally proximal relation along $C=\{p_1,\ldots,p_d\}$ (denoted by $RP_C^{[d]}(X,T)$) is introduced.
It turns out that for a minimal system, $RP_C^{[d]}(X,T)=Δ$ implies that $X$ is an almost one-to-one extension of $X_k$ for some $k\in\mathbb{N}$ only depending on a set of finite polynomials associated with $C$ and has zero entropy, where $X_k$ is the maximal $k$-step pro-nilfactor of $X$.
Particularly, when $C$ is a collection of linear polynomials, it is proved that $RP_C^{[d]}(X,T)=Δ$ implies $(X,T)$ is a $d$-step pro-nilsystem, which answers negatively a conjecture in \cite{5p}. The results are obtained by proving a refined saturation theorem for polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11251 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A refined saturation theorem for polynomials and applications Ye, Xiangdong Yu, Jiaqi Dynamical Systems For a dynamical system $(X,T)$, $d\in\mathbb{N}$ and distinct non-constant integral polynomials $p_1,\ldots, p_d$ vanishing at $0$, the notion of regionally proximal relation along $C=\{p_1,\ldots,p_d\}$ (denoted by $RP_C^{[d]}(X,T)$) is introduced. It turns out that for a minimal system, $RP_C^{[d]}(X,T)=Δ$ implies that $X$ is an almost one-to-one extension of $X_k$ for some $k\in\mathbb{N}$ only depending on a set of finite polynomials associated with $C$ and has zero entropy, where $X_k$ is the maximal $k$-step pro-nilfactor of $X$. Particularly, when $C$ is a collection of linear polynomials, it is proved that $RP_C^{[d]}(X,T)=Δ$ implies $(X,T)$ is a $d$-step pro-nilsystem, which answers negatively a conjecture in \cite{5p}. The results are obtained by proving a refined saturation theorem for polynomials. |
| title | A refined saturation theorem for polynomials and applications |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2405.11251 |