A refined saturation theorem for polynomials and applications

2024/05/18 by Xiangdong Ye, Jiaqi Yu, Ye, Xiangdong +1 · 1 citation
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2405.11251

openalex publication_date 2024/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a dynamical system (X,T), d∈ℕ and distinct non-constant integral polynomials p₁,…, p_d vanishing at 0, the notion of regionally proximal relation along C=\p₁,…,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∈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 \cite5p. The results are obtained by proving a refined saturation theorem for polynomials.

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