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 x ... x T and T x T-2 x ... x 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 (Xd), 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 showthat themaximal distal factor of N-d (X) is N-d (X-dis) with Xdis being themaximal 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 x T-2-saturated and is a Toeplitz minimal system.