论文标题
完全最小的系统中的多项式轨道
Polynomial orbits in totally minimal systems
论文作者
论文摘要
受Glasner，Huang，Shao，Weiss和Ye最近的作品的启发，我们证明了最小系统$（x，t）$的最大$ \ infty $ \ infty $ \ inty $ \ step pro-nilfactor $ x_ \ infty $是沿着一定意义上的拓扑特征。也就是说，我们表明，几乎一对一的修改为$π：x \ to x_ \ infty $，诱导的开放扩展$π^*：x^*\ to x_ \ infty^*$具有以下属性：对于任何$ d \ in \ in \ in \ mathbb {n} $ $ \ bigCap_ {i = 0}^dπ^*（v_i）\ neq \ emptySet $和任何不同的非稳定整数多项式$ p_i $ p_i $ with $ p_i（0）= 0 $ for $ i = 1，\ ldots，\ ldots，d $ t^{ - p_1（n）} v_1 \ cap \ ldots \ cap t^{ - p_d（n）} v_d \ neq \ emptyset $。整数多项式是多项式的，在整数上采用整数值。 作为应用程序，将获得以下结果：对于完全最小的系统$（x，t）$和整数多项式$ p_1，\ ldots，p_d $，如果每种非整数$ p_1的组合，\ ldots，\ ldots \ ldots \ ldots \ ldots \ ldots \ ldots \ ldots \ ldots \ ldots \ ldots \ ldots \ ldots均不恒定，那么$g_Δ$g_Δ$ subset $ subset $ upset $ x $ x $ x $ x x $ x x x $ x x x $ x x \ {（t^{p_1（n）} x，\ ldots，t^{p_d（n）} x）：n \ in \ mathbb {z} \} \]在每个$ x \ inω$中都是$ x^d $中的$ x^d $。
Inspired by the recent work of Glasner, Huang, Shao, Weiss and Ye, we prove that the maximal $\infty$-step pro-nilfactor $X_\infty$ 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 $π:X\to X_\infty$, the induced open extension $π^*:X^*\to X_\infty^*$ has the following property: for any $d\in \mathbb{N}$, any open subsets $V_0,V_1,\ldots,V_d$ of $X^*$ with $\bigcap_{i=0}^d π^*(V_i)\neq \emptyset$ and any distinct non-constant integer polynomials $p_i$ with $p_i(0)=0$ for $i=1,\ldots,d$, there exists some $n\in \mathbb{Z}$ such that $V_0\cap T^{-p_1(n)}V_1\cap \ldots \cap T^{-p_d(n)}V_d \neq \emptyset$. where an integer polynomial is the polynomial with rational coefficients taking integer values on the integers. As an application, the following result is obtained: for a totally minimal system $(X,T)$ and integer polynomials $p_1,\ldots,p_d$, if every non-trivial integer combination of $p_1,\ldots,p_d$ is not constant, then there is a dense $G_δ$ subset $Ω$ of $ X$ such that the set \[ \{(T^{p_1(n)}x,\ldots, T^{p_d(n)}x):n\in \mathbb{Z}\} \] is dense in $X^d$ for every $x\in Ω$.
