Regionally proximal relation of order d along arithmetic progressions and nilsystems
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September 2020 Vol. 63 No. 9: 1757–1776 https://doi.org/10.1007/s11425-019-1607-5
Regionally proximal relation of order d along arithmetic progressions and nilsystems Dedicated to Professor Shantao Liao
Eli Glasner1 , Wen Huang2 , Song Shao2 & Xiangdong Ye2,∗ 1School 2CAS
of Mathematics, Tel Aviv University, Tel Aviv 6997801, Israel; Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematics, University of Science and Technology of China, Hefei 230026, China
Email: [email protected], [email protected], [email protected], [email protected] Received May 22, 2019; accepted September 26, 2019; published online August 4, 2020
Abstract
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. Keywords MSC(2010)
regionally proximal relation, pro-nilsystem, discrete spectrum, equicontinuous factor 37B05, 37A99
Citation: Glasner E, Huang W, Shao S, et al. Regionally proximal relation of order d along arithmetic progressions and nilsystems. Sci China Math, 2020, 63: 1757–1776, https://doi.org/10.1007/s11425-019-1607-5
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Introduction
This paper is dedicated to the counterpart of the study of multiple ergodic averages in ergodic theory in the setting of topological dynamics. The regionally proximal relation of order d along arithmetic progressions, namely AP[d] for d ∈ N, is introduced and investigated. In some sense an equicontinuous system is the simplest system in topological dynamics. In the study of topological dynamics, one of the first problems was to characterize the equicontinuous structure relation Seq (X) of a system (X, T ), i.e., to find the smallest closed invariant equivalence relation R(X) on (X, T ) such that (X/R(X), T ) is equicontinuous. A natural candidate for R(X) is the so-called regionally proximal relation RP(X) introduced by Ellis and Gottschalk [10]. By the definition, RP(X) is closed, invariant, and reflexive, but not necessarily transitive. The problem was then to find conditions under which RP(X) is an equivalence relation. It turns out to be a difficult problem. Starting with Veech [34], * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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researchers, including MacMahon [31], Ellis and
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