Saturation of Product Systems and Applications

Jiahao Qiu^a, Hui Xu^b, Xiangdong Ye^a, Jiaqi Yu<sup>a

a</sup> School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China
^b Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

Abstract: Recently, Glasner, Huang, Shao, Weiss and Ye proved that the maximal infinite-step pro-nilfactor of a minimal system is a topological characteristic factor in a certain sense. In this paper, we extend their work to the product of finitely many minimal systems and determine the sharp order for a pro-nilfactor to qualify as a topological characteristic factor. As an application of our results, we characterize the complexity of most fibers within a minimal system from a topological perspective. As a corollary, we prove conjectures proposed by Huang et al. for $\mathbb Z$-actions.

Keywords: saturation theorems, independence, pro-nilsystems.

Received: May 5, 2025
Revised: June 2, 2025
Accepted: June 11, 2025

DOI: 10.4213/tm4473

 English version: Proceedings of the Steklov Institute of Mathematics, 2025, 330, 342–358

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