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] = 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.
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This work was supported by National Natural Science Foundation of China (Grant Nos. 11431012,11971455,11571335 and 11371339).
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Dedicated to Professor Shantao Liao
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Glasner, E., Huang, W., Shao, S. et al. Regionally proximal relation of order d along arithmetic progressions and nilsystems. Sci. China Math. 63, 1757–1776 (2020). https://doi.org/10.1007/s11425-019-1607-5
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DOI: https://doi.org/10.1007/s11425-019-1607-5