标题： 交换变换和极小$\mathbb{R}$-流的结构定理 显示英文标题

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

摘要： 本文中，我们发展了几个关于交换变换和极小的$\mathbb{R}$-流的结构定理。 具体来说，我们证明了如果$(X,S)$和$(X,T)$是由$S$和$T$共同确定的极小系统且是可交换的，则它们具有相同的高阶区域近似关系。 因此，$(X, S)$和$(X, T)$都有相同的增序列的拟幂零因子。 对于极小的$\mathbb{R}$-流，我们引入了高阶局部接近关系和幂零因子的概念，并证明幂零因子是极小的$\mathbb{R}$-流的特征因子，直到几乎一对一的扩张为止。 显示英文摘要

摘要： 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.