A refined saturation theorem for polynomials and applications

Ye, Xiangdong; Yu, Jiaqi

doi:10.1090/proc/17046

A refined saturation theorem for polynomials and applications
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by Xiangdong Ye and Jiaqi Yu;

Proc. Amer. Math. Soc. 153 (2025), 1077-1092

DOI: https://doi.org/10.1090/proc/17046

Published electronically: January 21, 2025

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Abstract:

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 $\mathbf {RP}_C^{[d]}(X,T)$) is introduced.

It turns out that for a minimal system, $\mathbf {RP}_C^{[d]}(X,T)=\Delta$ 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 $\mathbf {RP}_C^{[d]}(X,T)=\Delta$ implies $(X,T)$ is a $d$-step pro-nilsystem, which answers negatively a conjecture of Glasner, Huang, Shao, Weiss, and Ye [Topological characteristic factors and nilsystems, J. Eur. Math. Soc. (to appear), https://arxiv.org/abs/2006.12385, 2020]. The results are obtained by proving a refined saturation theorem for polynomials.

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