Topological characteristic factors and nilsystems

Abstract

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 $\pi :X\to X_{\infty }$, the induced open extension ${\pi }^{\ast }:X^{\ast }\to X_{\infty }^{\ast }$ has the following property: for $x$ in a dense $G_{\delta }$ subset of $X^{\ast }$, the orbit closure $L_x=\overline{\mathcal{O}}((x,\dots ,x),T\times T^2\times \cdots \times T^d)$ is $({\pi }^{\ast }{)}^{(d)}$-saturated, i.e., $L_x=(({\pi }^{\ast }{)}^{(d)}{)}^{-1}({\pi }^{\ast }{)}^{(d)}(L_x)$. 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 $T^{n_i}x\to x,T^{2n_i}x\to x,\dots ,T^{d{n}_i}x\to x$ for $x$ in a dense $G_{\delta }$ 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 $G_{\delta }$ subset of $X$; (3) for any $d\in \mathbb{N}$ and any minimal t.d.s. which is an open extension of its maximal distal factor, ${\mathbf{RP}}^{[d]}={\mathbf{AP}}^{[d]}$, where the former is the regionally proximal relation of order $d$ and the latter is the regionally proximal relation of order $d$ along arithmetic progressions.