[论文解读] Nil Bohr-sets and almost automorphy of higher order

Wen Huang, Song Shao+1 more|arXiv (Cornell University)|Jul 4, 2014

Mathematical Dynamics and Fractals参考文献 20被引用 3

一句话总结

本文通过尼尔系统（nilsystems）和广义多项式（generalized polynomials）建立了高阶Bohr集与高阶几乎自同构系统在拓扑动力系统中的深层联系。证明了Nil d Bohr⁰-集可通过广义多项式表征，并表明d步几乎自同构点由返回集和Nil d Bohr⁰-集表征，从而通过尼尔因子理论统一了高阶返回性与结构。

ABSTRACT

Two closely related topics: higher order Bohr sets and higher order almost automorphy are investigated in this paper. Both of them are related to nilsystems. In the first part, the problem which can be viewed as the higher order version of an old question concerning Bohr sets is studied: for any $d\in {\mathbb N}$ does the collection of $\{n\in {\mathbb Z}: S\cap (S-n)\cap\ldots\cap (S-dn) eq \emptyset\}$ with $S$ syndetic coincide with that of Nil$_d$ Bohr$_0$-sets? It is proved that Nil$_d$ Bohr$_0$-sets could be characterized via generalized polynomials, and applying this result one side of the problem is answered affirmatively: for any Nil$_d$ Bohr$_0$-set $A$, there exists a syndetic set $S$ such that $A\supset \{n\in {\mathbb Z}: S\cap (S-n)\cap\ldots\cap (S-dn) eq \emptyset\}.$ Moreover, it is shown that the answer of the other side of the problem can be deduced from some result by Bergelson-Host-Kra if modulo a set with zero density. In the second part, the notion of $d$-step almost automorphic systems with $d\in{\mathbb N}\cup\{\infty\}$ is introduced and investigated, which is the generalization of the classical almost automorphic ones. It is worth to mention that some results concerning higher order Bohr sets will be applied to the investigation. For a minimal topological dynamical system $(X,T)$ it is shown that the condition $x\in X$ is $d$-step almost automorphic can be characterized via various subsets of ${\mathbb Z}$ including the dual sets of $d$-step Poincaré and Birkhoff recurrence sets, and Nil$_d$ Bohr$_0$-sets. Moreover, it turns out that the condition $(x,y)\in X imes X$ is regionally proximal of order $d$ can also be characterized via various subsets of ${\mathbb Z}$.

研究动机与目标

• 解决一个关于syndetic集是否通过交集条件生成Nil d Bohr⁰-集的旧问题的高阶推广。
• 引入并研究d步几乎自同构系统这一概念，作为经典几乎自同构性的推广。
• 利用返回集和Nil d Bohr⁰-集表征d步几乎自同构点以及d阶区域接近对。
• 通过广义多项式表征，统一高阶返回性、尼尔系统与动力系统结构。

提出的方法

• 通过广义多项式表征Nil d Bohr⁰-集，表明其由d次广义多项式的水平集定义。
• 将Bergelson-Host-Kra定理模零密度集应用于分析高阶Bohr问题的逆方向。
• 利用Furstenberg对应原理将组合学中的返回集与拓扑动力系统联系起来。
• 通过d步尼尔系统的逆极限表征最大d步尼尔因子和d阶区域接近关系。
• 通过返回集与拓扑返回集定义并分析d步几乎自同构系统。
• 利用Ellis半群与紧致Hausdorff系统理论研究d阶区域接近对的结构。

实验结果

研究问题

• RQ1 每个Nil d Bohr⁰-集是否都可表示为形如{ n ∈ ℤ : S ∩ (S−n) ∩ … ∩ (S−dn) ≠ ∅ }的集合，其中S为syndetic集？
• RQ2 d步几乎自同构系统能否通过返回集（如d阶Poincaré集和Birkhoff集）表征？
• RQ3 Nil d Bohr⁰-集与极小系统中d阶区域接近关系之间有何关系？
• RQ4 广义多项式在表征高阶Bohr集的过程中起什么作用？
• RQ5 d步几乎自同构点的结构能否通过尼尔因子与返回集不变量完全描述？

主要发现

• Nil d Bohr⁰-集可表征为d次广义多项式的水平集，提供了新的代数描述。
• 对任意Nil d Bohr⁰-集A，存在一个syndetic集S，使得A包含{ n ∈ ℤ : S ∩ (S−n) ∩ … ∩ (S−dn) ≠ ∅ }，从而证实了高阶Bohr问题的一个方向。
• 高阶Bohr问题的逆方向在模零密度集下成立，由Bergelson-Host-Kra的结果推出。
• X中的点x是d步几乎自同构的，当且仅当其轨道闭包包含于d步返回点集的闭包中。
• d阶区域接近关系由d重返回集与Nil d Bohr⁰-集的交集表征。
• 极小系统(X,T)的最大d步尼尔因子同构于X / RP^{[d]}，且该因子捕捉了d步几乎自同构结构。