TY - JOUR

T1 - Regionally proximal relation of order d along arithmetic progressions and nilsystems

AU - Glasner, Eli

AU - Huang, Wen

AU - Shao, Song

AU - Ye, Xiangdong

N1 - Publisher Copyright:
© 2020, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - 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] = Δ, 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.

AB - 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] = Δ, 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.

KW - 37A99

KW - 37B05

KW - discrete spectrum

KW - equicontinuous factor

KW - pro-nilsystem

KW - regionally proximal relation

UR - http://www.scopus.com/inward/record.url?scp=85089252699&partnerID=8YFLogxK

U2 - 10.1007/s11425-019-1607-5

DO - 10.1007/s11425-019-1607-5

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AN - SCOPUS:85089252699

SN - 1674-7283

VL - 63

SP - 1757

EP - 1776

JO - Science China Mathematics

JF - Science China Mathematics

IS - 9

ER -