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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References
Auslander J. Minimal Flows and Their Extensions. North-Holland Mathematics Studies, vol. 153. Amsterdam: North-Holland, 1988
Blanchard F. Fully positive topological entropy and topological mixing: Symbolic dynamics and its applications. Contemp Math, 1992, 135: 95–105
Blanchard F. A disjointness theorem involving topological entropy. Bull Soc Math France, 1993, 121: 465–478
Blanchard F, Host B, Ruette S. Asymptotic pairs in positive-entropy systems. Ergodic Theory Dynam Systems, 2002, 22: 671–686
Bronstein I. Extensions of Minimal Transformation Groups. Netherlands: Springer, 1979
Cai F, Shao S. Topological characteristic factors along cubes of minimal systems. Discrete Contin Dyn Syst, 2019, 39: 5301–5317
de Vries J. Elements of Topological Dynamics. Dordrecht: Kluwer Academic Publishers, 1993
Dong P, Donoso S, Maass S, et al. Infinite-step nilsystems, independence and complexity. Ergodic Theory Dynam Systems, 2013, 33: 118–143
Ellis R, Glasner S, Shapiro L. Proximal-isometric (PJ) flows. Adv Math, 1975, 17: 213–260
Ellis R, Gottschalk W. Homomorphisms of transformation groups. Trans Amer Math Soc, 1960, 94: 258–271
Ellis R, Keynes H. A characterization of the equicontinuous structure relation. Trans Amer Math Soc, 1971, 161: 171–183
Furstenberg H. Strict ergodicity and transformation of the torus. Amer J Math, 1961, 83: 573–601
Furstenberg H. The structure of distal flows. Amer J Math, 1963, 85: 477–515
Furstenberg H. Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions. J Anal Math, 1977, 31: 204–256
Furstenberg H. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton: Princeton University Press, 1981
Furstenber H, Weiss B. A mean ergodic theorem for \({1 \over N}\sum\nolimits_{n = 1}^N {f\left({{T^n}x} \right)g\left({T{n^2}x} \right)} \). In: Convergence in Ergodic Theory and Probability. Ohio State University Mathematical Research Institute Publications, vol. 5. Berlin: de Gruyter, 1996, 193–227
Glasner E. Topological ergodic decompositions and applications to products of powers of a minimal transformation. J Anal Math, 1994, 64: 241–262
Glasner E, Gutman Y, Ye X. Higher order regionally proximal equivalence relations for general minimal group actions. Adv Math, 2018, 333: 1004–1041
Glasner E, Weiss B. Quasi-factors of zero-entropy systems. J Amer Math Soc, 1995, 8: 665–686
Glasner S. Relatively invariant measures. Pacific J Math, 1975, 58: 393–410
Host B, Kra B. Nonconventional ergodic averages and nilmanifolds. Ann of Math (2), 2005, 161: 397–488
Host B, Kra B. Nilpotent Structures in Ergodic Theory. Mathematical Surveys and Monographs, vol. 236. Providence: Amer Math Soc, 2018
Host B, Kra B, Maass A. Nilsequences and a structure theorem for topological dynamical systems. Adv Math, 2010, 224: 103–129
Host B, Kra B, Maass A. Variations on topological recurrence. Monatsh Math, 2016, 179: 57–89
Huang W, Li H, Ye X. Family-independence for topological and measurable dynamics. Trans Amer Math Soc, 2012, 364: 5209–5242
Huang W, Shao S, Ye X. Nil Bohr-sets and almost automorphy of higher order. Mem Amer Math Soc, 2016, 241: 1143
Huang W, Ye X. An explicit scattering, non-weakly mixing example and weak disjointness. Nonlinearity, 2002, 15: 849–862
Huang W, Ye X. A local variational relation and applications. Israel J Math, 2006, 151: 237–280
Kerr D, Li H. Independence in topological and C*-dynamics. Math Ann, 2007, 338: 869–926
Lehrer E. Topological mixing and uniquely ergodic systems. Israel J Math, 1987, 57: 239–255
McMahon D. Relativized weak disjointness and relatively invariant measures. Trans Amer Math Soc, 1978, 236: 225–237
Qiao Y X. Topological complexity, minimality and systems of order two on torus. Sci China Math, 2016, 59: 503–514
Shao S, Ye X. Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence. Adv Math, 2012, 231: 1786–1817
Veech W. The equicontinuous structure relation for minimal Abelian transformation groups. Amer J Math, 1968, 90: 723–732
Ziegler T. Universal characteristic factors and Furstenberg averages. J Amer Math Soc, 2007, 20: 53–97
Zygmund A. Trigonormetric Series, 2nd ed. Cambridge: Cambridge University Press, 1959
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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