论文标题
拓扑特征因素和沿算术进展的独立性
Topological characteristic factors and independence along arithmetic progressions
论文作者
论文摘要
令$π：（x，t）\ rightarrow（y，t）$为拓扑动力学的因子图，而$ d \ in {\ mathbb {n}} $中的$ d \。 $（y，t）$据说是一个$ d $ step的拓扑特征因素，如果存在密集的$g_Δ$设置$ x_0 $ x $的$ x $，以便对于x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ the orbit lib the orbit line $ \ overline {\ nathcal o}（\ nathcal o} $π\ times \ ldots \ timesπ$（$ d $ times）饱和。 1994年，埃利·格拉斯纳（Eli Glasner）研究了最小系统的拓扑特征因子。例如，结果表明，对于远端最小系统，其最大的derford $ d-1 $远端因素是其$ d $ step-step的拓扑特征因子。在本文中，我们将格拉斯纳的工作推广到有限的许多最小系统的产品系统，并提供其相对版本。为了证明这些结果，我们需要处理$（x，t^m）$ for $ m \ in {\ mathbb {n}} $。我们将研究$（x，t^m）$的结构定理。我们表明，尽管对于最小的系统$（x，t）$和$ m \ in {\ mathbb {n}} $，$（x，t^m）$可能不是最小的，但是我们仍然可以使用$（x，t^m）$的pi-tower，实际上与$（x，x，x，t）$的pi tower相同。我们提供了一些结果的应用。例如，我们表明，如果最小的系统沿算法$ d $的算术进程没有非平凡的独立对，则最多可以达到规范定义的近端扩展，它是订单$ d $的pi;如果最小的系统$（x，t）$具有非平凡的$ d $ step拓扑特征因素，则存在``许多''$δ$交易式订单$ d $。
Let $π: (X,T)\rightarrow (Y,T)$ be a factor map of topological dynamics and $d\in {\mathbb {N}}$. $(Y,T)$ is said to be a $d$-step topological characteristic factor if there exists a dense $G_δ$ set $X_0$ of $X$ such that for each $x\in X_0$ the orbit closure $\overline{\mathcal O}((x, \ldots,x), T\times T^2\times \ldots \times T^d)$ is $π\times \ldots \times π$ ($d$ times) saturated. In 1994 Eli Glasner studied the topological characteristic factor for minimal systems. For example, it is shown that for a distal minimal system, its largest distal factor of order $d-1$ is its $d$-step topological characteristic factor. In this paper, we generalize Glasner's work to the product system of finitely many minimal systems and give its relative version. To prove these results, we need to deal with $(X,T^m)$ for $m\in {\mathbb {N}}$. We will study the structure theorem of $(X,T^m)$. We show that though for a minimal system $(X,T)$ and $m\in {\mathbb {N}}$, $(X,T^m)$ may not be minimal, but we still can have PI-tower for $(X,T^m)$ and in fact it looks the same as the PI tower of $(X,T)$. We give some applications of the results developed. For example, we show that if a minimal system has no nontrivial independent pair along arithmetic progressions of order $d$, then up to a canonically defined proximal extension, it is PI of order $d$; if a minimal system $(X,T)$ has a nontrivial $d$-step topological characteristic factor, then there exist ``many'' $Δ$-transitive sets of order $d$.
