On structure theorems and non-saturated examples

TL;DR

This paper investigates the structure of certain minimal systems derived from a base system, establishing their maximal distal factors and structure theorems, and provides a non-saturated Toeplitz example.

Contribution

It extends the understanding of the structure of $N_d(X)$ systems, showing they share the same structure theorem as the original system and constructing a non-saturated Toeplitz example.

Findings

Maximal distal factor of $N_d(X)$ is $N_d(X_{dis})$.

$(N_{d}(X), ext{group})$ has the same structure theorem as $(X,T)$.

Constructed a non-saturated Toeplitz minimal system.

Abstract

For any minimal system $(X,T)$ and $d\geq 1$ there is an associated minimal system $(N_{d}(X), \mathcal{G}_{d}(T))$, where $\mathcal{G}_{d}(T)$ is the group generated by $T\times\cdots\times T$ and $T\times T^2\times\cdots\times T^{d}$ and $N_{d}(X)$ is the orbit closure of the diagonal under $\mathcal{G}_{d}(T)$. It is known that the maximal $d$-step pro-nilfactor of $N_d(X)$ is $N_d(X_d)$, where $X_d$ is the maximal $d$-step pro-nilfactor of $X$. In this paper, we further study the structure of $N_d(X)$. We show that the maximal distal factor of $N_d(X)$ is $N_d(X_{dis})$ with $X_{dis}$ being the maximal distal factor of $X$, and prove that as minimal systems $(N_{d}(X), \mathcal{G}_{d}(T))$ has the same structure theorem as $(X,T)$. In addition, a non-saturated metric example $(X,T)$ is constructed, which is not $T\times T^2$-saturated and is a Toeplitz minimal system.