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 system $(N_{d}(X), {\mathcal {G}}_{d}(T))$ 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.