论文标题
拓扑特征因素和nilsystems
Topological characteristic factors and nilsystems
论文作者
论文摘要
我们证明，最小动态系统$（x，t）$的最大无限步骤pro-nilfactor $ x_ \ infty $在某种意义上是拓扑特征因素。也就是说，我们表明，几乎一对一的修改$π：x \ rightarrow x_ \ infty $，诱导的开放式扩展$π^*：x^*\ rightArrow x^*_ \ infty $具有以下属性：$ x $在密集的$g_δ$中，$ x^$，$ x^*$，$ x^*$， $ l_x = \ overline {\ mathcal {o}}（（（x，x，x，x，\ ldots，x），t \ times t^2 \ times \ times \ ldots \ times t^d）$ is $（π^*）^*）^{（d）} $ - $ l_x =（（π^*）^{（d）}）^{ - 1}（π^*）^{（d）}（l_x）$。 使用以上事实得出的结果，我们能够回答几个空旷的问题：（1）如果$（x，t^k）$对于某些$ k \ ge 2 $是最小的，那么对于任何$ d \ in {\ mathbb n} $中的任何$ d \ （\ text {mod} \ k）$这样，以至于$ t^{n_i} x \ rightarrow x，t^{2n_i} x \ rightArol x，\ ldots，t^{dn_i} x \ rightArrow x $ for $ x $ for $ x $ for $ x $ in agenense $g_δ$ of $g_δ$ of $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ （2）如果$（x，t）$完全很小，则$ \ {t^{n^2} x：n \ in {\ mathbb z} \} $在$ x $中in $ x $ in $ x $中的$ x $在$g_Δ$ subset中$ x $ in $ x $ $ x $; x $; （3）对于任何$ d \ in \ mathbb n $和任何最小系统，这是其最大远端因素的开放式扩展，$ {\ bf rp}^{[d]} = {\ bf ap}^{[d ap}^{[d]} $，后者是后者的区域近端关系$ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d c。
We prove that the maximal infinite step pro-nilfactor $X_\infty$ of a minimal dynamical system $(X,T)$ is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of $π:X \rightarrow X_\infty$, the induced open extension $π^*:X^* \rightarrow X^*_\infty$ has the following property: for $x$ in a dense $G_δ$ set of $X^*$, the orbit closure $L_x=\overline{\mathcal{O}}((x,x,\ldots,x), T\times T^2\times \ldots \times T^d)$ is $(π^*)^{(d)}$-saturated, i.e. $L_x=((π^*)^{(d)})^{-1}(π^*)^{(d)}(L_x)$. Using results derived from the above fact, we are able to answer several open questions: (1) if $(X,T^k)$ is minimal for some $k\ge 2$, then for any $d\in {\mathbb N}$ and any $0\le j<k$ there is a sequence $\{n_i\}$ of $\mathbb Z$ with $n_i\equiv j\ (\text{mod}\ k)$ such that $T^{n_i}x\rightarrow x, T^{2n_i}x\rightarrow x, \ldots, T^{dn_i}x\rightarrow x$ for $x$ in a dense $G_δ$ subset of $X$; (2) if $(X,T)$ is totally minimal, then $\{T^{n^2}x:n\in {\mathbb Z}\}$ is dense in $X$ for $x$ in a dense $G_δ$ subset of $X$; (3) for any $d\in\mathbb N$ and any minimal system, which is an open extension of its maximal distal factor, ${\bf RP}^{[d]}={\bf AP}^{[d]}$, where the latter is the regionally proximal relation of order $d$ along arithmetic progressions.
