Polynomial orbits in totally minimal systems

Abstract

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 $\pi :X\to X_{\infty }$, the induced open extension ${\pi }^{\ast }:X^{\ast }\to X_{\infty }^{\ast }$ has the following property: for any $d\in \mathbb{N}$, any open subsets $V_0,V_1,\dots ,V_d$ of $X^{\ast }$ with ${\bigcap }_{i=0}^d{\pi }^{\ast }(V_i)\ne \varnothing$ and any distinct non-constant integer polynomials $p_i$ with $p_i(0)=0$ for $i=1,\dots ,d$, there exists some $n\in \mathbb{Z}$ such that $V_0\cap T^{-p_1(n)}{V}_1\cap \dots \cap T^{-p_d(n)}{V}_d\ne \varnothing$. 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 $p_1,\dots ,p_d$, if every non-trivial integer combination of $p_1,\dots ,p_d$ is not constant, then there is a dense $G_{\delta }$ subset $\Omega$ 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 $x\in \Omega$.

arXiv admin note: text overlap with arXiv:2006.12385, arXiv:1407.1179 by other authors