On structure theorems and non-saturated examples

Abstract

For any minimal system $(X,T)$ and $d\ge 1$ 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.