Saturation of Product Systems and Applications

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.