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

Song Shao; Hui Xu

doi:10.3934/dcds.2026045

Article Contents

2026, Volume 52: 70-114. Doi: 10.3934/dcds.2026045

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

Song Shao^1, and
Hui Xu^2, ,

1.
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
2.
Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

^*Corresponding author: Hui Xu
^*Corresponding author: Hui Xu

Received: May 8, 2025

Revised: February 22, 2026

Early access: March 2, 2026

Published online: March 2, 2026

This research is supported by National Key R&D Program of China (No. 2024YFA1013601, 2024YFA1013600), and National Natural Science Foundation of China (12426201, 12371196, 12201599).

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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 share 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.

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Mathematics Subject Classification: Primary: 37B05; Secondary: 37B20, 37A17.

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