On Structure Theorems and Non-saturated Examples

For any minimal system ( X , T ) and d≥ 1 , there is an associated minimal system (N_d(X), 𝒢_d(T)) , where 𝒢_d(T) is the group generated by T×⋯× T and T× T^2×⋯× T^d , and N_d(X) is the orbit closure of the diagonal under 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), 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× T^2 -saturated and is a Toeplitz minimal system.