# Topological characteristic factors and nilsystems

### Eli Glasner

Tel Aviv University, Israel### Wen Huang

University of Science and Technology of China, Hefei, China### Song Shao

University of Science and Technology of China, Hefei, China### Benjamin Weiss

The Hebrew University of Jerusalem, Israel### Xiangdong Ye

University of Science and Technology of China, Hefei, China

## Abstract

We prove that the maximal infinite step pro-nilfactor $X_{∞}$ 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 $π:X→X_{∞}$, the induced open extension $π_{∗}:X_{∗}→X_{∞}$ has the following property: for $x$ in a dense $G_{δ}$ subset of $X_{∗}$, the orbit closure $L_{x}=O((x,…,x),T×T_{2}×⋯×T_{d})$ is $(π_{∗})_{(d)}$-saturated, i.e., $L_{x}=((π_{∗})_{(d)})_{−1}(π_{∗})_{(d)}(L_{x})$. Using results derived from the above fact, we are able to answer several open questions: (1) if $(X,T_{k})$ is minimal for some $k≥2$, then for any $d∈N$ and any $0≤j<k$ there is a sequence ${n_{i}}$ of $Z$ with $n_{i}≡j(modk)$ such that $T_{n_{i}}x→x,T_{2n_{i}}x→x,…,T_{dn_{i}}x→x$ for $x$ in a dense $G_{δ}$ subset of $X$; (2) if $(X,T)$ is totally minimal, then ${T_{n_{2}}x:n∈Z}$ is dense in $X$ for $x$ in a dense $G_{δ}$ subset of $X$; (3) for any $d∈N$ and any minimal t.d.s. which is an open extension of its maximal distal factor, $RP_{[d]}=AP_{[d]}$, 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.

## Cite this article

Eli Glasner, Wen Huang, Song Shao, Benjamin Weiss, Xiangdong Ye, Topological characteristic factors and nilsystems. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1379