Polynomial orbits in totally minimal systems

TL;DR

This paper establishes that the maximal pro-nilfactor of a minimal system serves as a topological characteristic factor along polynomials, leading to new density results for polynomial orbits in totally minimal systems.

Contribution

It proves the maximal pro-nilfactor is a topological characteristic factor along polynomials and derives density results for polynomial orbits in totally minimal systems.

Findings

Maximal pro-nilfactor acts as a topological characteristic factor along polynomials.

Existence of points with dense polynomial orbits in totally minimal systems.

Extension properties ensure polynomial recurrence in open sets.

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 $\pi^*:X^*\to X_\infty^*$ has the following property: for any $d\in \mathbb{N}$, any open subsets $V_0,V_1,\ldots,V_d$ of $X^*$ with $\bigcap_{i=0}^d \pi^*(V_i)\neq \emptyset$ and any distinct non-constant integer polynomials $p_i$ with $p_i(0)=0$ for $i=1,\ldots,d$, there exists some $n\in \mathbb{Z}$ such that $V_0\cap T^{-p_1(n)}V_1\cap \ldots \cap T^{-p_d(n)}V_d \neq \emptyset$. 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 $p_1,\ldots,p_d$, if every non-trivial integer combination of $p_1,\ldots,p_d$ 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 $x\in \Omega$.