论文标题
拓扑动力学系统中算术进程空间的亲能剂
Pro-nilfactors of the space of arithmetic progressions in topological dynamical systems
论文作者
论文摘要
对于拓扑动力系统，$（x，t）$，$ l \ in \ mathbb {n} $和$ x \ in x $，让$ n_l（x）$和$ l_x^l（x）$是对角点$的轨道关闭$（x，x，x，x，x，x，\ ldots，x）$ time $ time $ time $ time $ time $ time $ time $ time $ n ocation $ {分别$ \ \ $ \ MATHCAL {G} _ {l} $由$ t \ times t \ times t \ times \ ldots \ times t $（$ l $ times）和$τ_l= t \ t \ times t^2 \ times t^2 \ times \ ldots \ ldots \ ldots \ ldots \ ldots \ ldots \ ldots \ times t^l $。在本文中，我们表明，对于最小的系统$（x，t）$和$ l \ in \ mathbb {n} $，最大$ d $ -d $ -Step pro-nilfactor的$（n_l（x），\ Mathcal {g} _ {l} _ {l} _ {l}） $π_d：x \ to x/\ mathbf {rp}^{[[d]} = x_d，d \ in \ mathbb {n} $是因子映射，$ \ mathbf {rp}^{[d]} $是订单$ d $的区域性关系。 同时，当$（x，t）$是一个最小的nilsystem时，我们还计算几乎每$ x $ w.r.t.的$（l_x^l（x），τ_l）$的pro-nilfactors $。 HAAR措施。特别是，存在最小的$ 2 $ -Step nilsystem $（y，t）$和一个可数的套装$ω\ subset y $，因此对于$ y \ in y \ backslashω$，最大的等效因子为$（l_y^2（y），τ_2）$（τ_2）$ $（l_ {π_1（y）}^2（y_ {1}），τ_2）$。
For a topological dynamical system $(X, T)$, $l\in\mathbb{N}$ and $x\in X$, let $N_l(X)$ and $L_x^l(X)$ be the orbit closures of the diagonal point $(x,x,\ldots,x)$ ($l $ times) under the actions $\mathcal{G}_{l}$ and $τ_l $ respectively, where $\mathcal{G}_{l}$ is generated by $T\times T\times \ldots \times T$ ($l $ times) and $τ_l=T\times T^2\times \ldots \times T^l$. In this paper, we show that for a minimal system $(X,T)$ and $l\in \mathbb{N}$, the maximal $d$-step pro-nilfactor of $(N_l(X),\mathcal{G}_{l})$ is $(N_l(X_d),\mathcal{G}_{l})$, where $π_d:X\to X/\mathbf{RP}^{[d]}=X_d,d\in \mathbb{N}$ is the factor map and $\mathbf{RP}^{[d]}$ is the regionally proximal relation of order $d$. Meanwhile, when $(X,T)$ is a minimal nilsystem, we also calculate the pro-nilfactors of $(L_x^l(X),τ_l)$ for almost every $x$ w.r.t. the Haar measure. In particular, there exists a minimal $2$-step nilsystem $(Y,T)$ and a countable set $Ω\subset Y$ such that for $y\in Y\backslash Ω$ the maximal equicontinuous factor of $(L_y^2(Y),τ_2)$ is not $(L_{π_1(y)}^2(Y_{1}),τ_2)$.
