Regionally proximal relation of order d along arithmetic progressions and nilsystems
Regionally proximal relation of order d along arithmetic progressions and nilsystems
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DOI：
10.1007/s11425-019-1607-5
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发表时间：
2020
影响因子：
1.4
通讯作者：
Ye Xiangdong
中科院分区：
文献类型：
--
作者：
Glasner Eli;Huang Wen;Shao Song;Ye Xiangdong
The regionally proximal relation of order d along arithmetic progressions, namely AP~([d]) for d ∈ N, is introduced and investigated. It turns out that if (X, T ) is a topological dynamical system with AP~([d]) = ∆, then each ergodic measure of (X, T ) is isomorphic to a d-step pro-nilsystem, and thus (X, T ) has zero entropy. Moreover, it is shown that if (X, T ) is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic, then AP~([d]) = RP~([d]) for each d ∈ N . It follows that for a minimal ∞ -pro-nilsystem, AP~([d]) = RP~([d]) for each d ∈ N . An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.
