Polynomial orbits in totally minimal systems
Polynomial orbits in totally minimal systems
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DOI：
10.1016/j.aim.2023.109260
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发表时间：
2023-11
影响因子：
1.7
通讯作者：
Jiahao Qiu
中科院分区：
文献类型：
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作者：
Jiahao Qiu
本文受Reynner，Huang，Shao，韦斯和Ye [14]的启发，证明了极小系统（X，T）的极大∞步原零因子X∞在一定意义下是沿沿着多项式的拓扑特征因子.也就是说，我们证明了通过π：X→ X∞的几乎一对一的修改，导出的开扩张π → X∞具有以下性质：对任意d∈ N，X的任意开子集V0，V1，.，Vd，其中n i= 0 d π（Vi）n，以及任意不同的非常数整数多项式pi，其中pi（0）= 0，i= 1，.，d，存在某个n∈ Z使得V 0。作为应用，得到了如下结果：对于完全极小系（X，T）和整数多项式p1，.，pd，如果p1，.，pd的每个非平凡整数组合都不是常数，则存在X的一个稠密G δ子集Ω，使得集合{（Tp 1（n）x，.，Tpd（n）x）：n∈ Z}在Xd中对每个x∈ Ω都是稠密的.
Inspired by the recent work of Glasner, Huang, Shao, Weiss and Ye [14], we prove that the maximal∞-step pro-nilfactor X∞ of a minimal system (X, T) is the topological characteristic factor along polynomials in a certain sense. Namely, we show that by an almost one to one modification of π: X→ X∞, the induced open extension π⁎: X⁎→ X∞⁎ has the following property: for any d∈ N, any open subsets V 0, V 1,…, V d of X⁎ with⋂ i= 0 d π⁎(V i)≠∅ and any distinct non-constant integer polynomials p i with p i (0)= 0 for i= 1,…, d, there exists some n∈ Z such that V 0∩ T− p 1 (n) V 1∩…∩ T− p d (n) V d≠∅. As an application, the following result is obtained: for a totally minimal system (X, T) and integer polynomials p 1,…, p d, if every non-trivial integer combination of p 1,…, p d is not constant, then there is a dense G δ subset Ω of X such that the set {(T p 1 (n) x,…, T p d (n) x): n∈ Z} is dense in X d for every x∈ Ω.