Structure theorems of commuting transformations and minimal $\mathbb{R}$-flows

TL;DR

This paper establishes structural theorems for commuting transformations and minimal real flows, revealing shared proximal relations and nilfactors, advancing understanding of their dynamical properties.

Contribution

It introduces higher-order regionally proximal relations and nilfactors for minimal $ extbf{R}$-flows, showing they are characteristic factors up to extensions.

Findings

Commuting minimal systems share the same higher-order proximal relations.

Minimal $ extbf{R}$-flows have nilfactors as characteristic factors.

Shared structure of pro-nilfactors in commuting systems.

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.