On structure theorems and non-saturated examples

2022/01/01 by Qinqi Wu, Hui Xu, Wu, Qinqi +3
Mathematics · #37B05 #54H20 #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2201.00152

openalex publication_date 2022/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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× T²×⋯× 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 systems (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× T²-saturated and is a Toeplitz minimal system.

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