Structure theorems of commuting transformations and minimal $\mathbb{R}$-flows
Song Shao, Hui Xu
TL;DR
The paper develops structure theorems for commuting transformations and minimal $\mathbb{R}$-flows, establishing that minimal systems with commuting actions share identical higher-order regionally proximal relations and pro-nilfactor hierarchies, i.e., $RP^{[d]}$ and $X_d=X/\mathbf{RP}^{[d]}$ coincide across actions. For $\mathbb{R}$-flows, it introduces higher-order proximal relations and nilfactors, proving nilfactors are topological characteristic factors up to almost one-to-one extensions, with a robust suspension-compatibility framework. The results unify discrete and continuous-time dynamics, leveraging dynamical cubes $Q^{[d]}$, $N_d$-structures, and open $O$-diagram constructions to reveal a common pro-nilpotent underlying structure. This advances the understanding of multiple recurrence and factorization in topological dynamics, providing tools for identifying characteristic factors and simplifying complex limit behaviors in both discrete and continuous time.
Abstract
In this paper, we develop several structure theorems concerning commuting transformations and minimal $\mathbb{R}$-flows. Specifically, we show that if $(X,S)$, $(X,T)$ are minimal systems with $S$ and $T$ being commutative, then they possess an identical higher-order regionally proximal relation. Consequently, both $(X, S)$ and $(X, T)$ share the same increasing sequence of pro-nilfactors. For minimal $\mathbb{R}$-flows, we introduce the concept of higher-order regionally proximal relations and nilfactors, and establish that nilfactors are characteristic factors for minimal $\mathbb{R}$-flows, up to almost one to one extensions.