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Higher order regionally proximal equivalence relations for general minimal group actions. (English) Zbl 1394.37016
Let \((G, X)\) denote a topological dynamical system, where \(G\) is a (Hausdorff) topological group and \(X\) is a compact Hausdorff space. The authors present a new definition, the nilpotent regionally proximal relations of order \(d\) (NRP\(^{[d]}(X)\) for short) \((d \in \mathbb{N})\), defined for general group actions \((G, X)\). They prove several results that play a key role in the paper. The authors prove that the nilpotent higher-order regionally proximal relations are equivalence relations for general minimal group actions. The authors show that the nilpotent regionally proximal equivalence relations lift through dynamical morphisms between minimal systems. They study the structure of systems whose nilpotent regionally proximal equivalence relation of order \(d\) is trivial. The relation between the classical regionally proximal relation and the nilpotent regionally proximal equivalence relation of order one is investigated. S. Shao and X. Ye [Adv. Math. 231, No. 3–4, 1786–1817 (2012; Zbl 1286.37010)] showed that RP\(^{[d]}(X)\) is an equivalence relation for minimal actions by abelian groups. The main result of the paper is the following theorem: let \((G, X)\) be a minimal topological dynamical system, then NRP\(^{[d]}(X)\) \((d \geq 1)\) is a closed \(G\)-invariant equivalence relation. This result is surprising as the regionally proximal relation RP\((X)\) is known not to be an equivalence relation for some (non-amenable) group actions. A different higher-order generalization of the classical regionally proximal relation for arbitrary group actions is presented. The authors exhibit an example related to the main result.

MSC:
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B10 Symbolic dynamics
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37B20 Notions of recurrence and recurrent behavior in dynamical systems
54H20 Topological dynamics (MSC2010)
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References:
[1] Akin, E., The general topology of dynamical systems, vol. 1, (2010), American Mathematical Society
[2] E. Akin, private communication.
[3] Akin, E.; Auslander, J.; Glasner, E., The topological dynamics of Ellis actions, (2008), American Mathematical Society · Zbl 1152.54026
[4] Antolín Camarena, O.; Szegedy, B., Nilspaces, nilmanifolds and their morphisms, (2012), preprint
[5] Auslander, J., Minimal flows and their extensions, North-Holland Mathematics Studies, vol. 153, (1988), North-Holland Publishing Co. Amsterdam, Notas de Matemática (Mathematical Notes), vol. 122 · Zbl 0654.54027
[6] Auslander, L.; Hahn, F.; Green, L.; Markus, L.; Massey, W., Flows on homogeneous spaces, vol. 53, (1963), Princeton University Press
[7] Candela, P., Notes on compact nilspaces, Discrete Anal., 16, (2017) · Zbl 1404.57053
[8] Candela, P., Notes on nilspaces - algebraic aspects, Discrete Anal., 15, (2017) · Zbl 1404.11011
[9] de Vries, J., Elements of topological dynamics, Mathematics and Its Applications, vol. 257, (1993), Kluwer Academic Publishers Group Dordrecht · Zbl 0783.54035
[10] Ellis, D. B.; Ellis, R., Automorphisms and equivalence relations in topological dynamics, vol. 412, (2014), Cambridge University Press · Zbl 1322.37001
[11] Ellis, R.; Gottschalk, W. H., Homomorphisms of transformation groups, Trans. Amer. Math. Soc., 94, 258-271, (1960) · Zbl 0094.17401
[12] Ellis, R.; Keynes, H., A characterization of the equicontinuous structure relation, Trans. Amer. Math. Soc., 161, 171-183, (1971) · Zbl 0233.54023
[13] Ellis, R.; Glasner, S.; Shapiro, L., Proximal-isometric (P J) flows, Adv. Math., 17, 3, 213-260, (1975) · Zbl 0304.54039
[14] Furstenberg, H., Strict ergodicity and transformation of the torus, Amer. J. Math., 83, 4, 573-601, (1961) · Zbl 0178.38404
[15] Glasner, S., Proximal flows, (1976), Springer · Zbl 0322.54017
[16] Glasner, E., Topological ergodic decompositions and applications to products of powers of a minimal transformation, J. Anal. Math., 64, 241-262, (1994) · Zbl 0855.54048
[17] Glasner, E., Ergodic theory via joinings, Mathematical Surveys and Monographs, vol. 101, (2003), American Mathematical Society Providence, RI · Zbl 1038.37002
[18] Glasner, E.; Megrelishvili, M.; Uspenskij, V. V., On metrizable enveloping semigroups, Israel J. Math., 164, 1, 317-332, (2008) · Zbl 1147.22003
[19] Green, B.; Tao, T., Linear equations in primes, Ann. of Math. (2), 171, 3, 1753-1850, (2010) · Zbl 1242.11071
[20] Gutman, Y.; Manners, F.; Varjú, P. P., The structure theory of nilspaces I, J. Anal. Math., (2018), in press
[21] Gutman, Y.; Manners, F.; Varjú, P. P., The structure theory of nilspaces III: inverse limit representations and topological dynamics, (2016), preprint
[22] Gutman, Y.; Manners, F.; Varjú, P. P., The structure theory of nilspaces II: representation as nilmanifolds, Trans. Amer. Math. Soc., (2018), in press · Zbl 1440.37017
[23] Host, B.; Kra, B., Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2), 161, 1, 397-488, (2005) · Zbl 1077.37002
[24] Host, B.; Kra, B., Parallelepipeds, nilpotent groups and Gowers norms, Bull. Soc. Math. France, 136, 3, 405-437, (2008) · Zbl 1189.11006
[25] Host, B.; Kra, B.; Maass, A., Nilsequences and a structure theorem for topological dynamical systems, Adv. Math., 224, 1, 103-129, (2010) · Zbl 1203.37022
[26] Magnus, W.; Karrass, A.; Solitar, D., Combinatorial group theory: presentations of groups in terms of generators and relations, (1966), Interscience Publishers [John Wiley & Sons, Inc.] New York-London-Sydney · Zbl 0138.25604
[27] McMahon, D., Weak mixing and a note on a structure theorem for minimal transformation groups, Illinois J. Math., 20, 2, 186-197, (1976) · Zbl 0316.54037
[28] McMahon, D. C., Relativized weak disjointness and relatively invariant measures, Trans. Amer. Math. Soc., 236, 225-237, (1978) · Zbl 0388.54030
[29] Parry, W., Dynamical systems on nilmanifolds, Bull. Lond. Math. Soc., 2, 1, 37-40, (1970) · Zbl 0194.05601
[30] Parry, W., Topics in ergodic theory, Cambridge Tracts in Mathematics, vol. 75, (1981), Cambridge University Press Cambridge-New York · Zbl 0449.28016
[31] Sepanski, M. R., Compact Lie groups, Graduate Texts in Mathematics, vol. 235, (2007), Springer New York · Zbl 1246.22001
[32] Shao, S.; Ye, X., Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence, Adv. Math., 231, 3-4, 1786-1817, (2012) · Zbl 1286.37010
[33] Szegedy, B., On higher order Fourier analysis, (2012), preprint
[34] Tao, T., A nonstandard analysis proof of Szemeredi’s theorem, (20 July, 2015), Blog post
[35] Tu, S.; Ye, X., Dynamical parallelepipeds in minimal systems, J. Dynam. Differential Equations, 25, 3, 765-776, (2013) · Zbl 1278.37016
[36] Veech, W. A., The equicontinuous structure relation for minimal abelian transformation groups, Amer. J. Math., 723-732, (1968) · Zbl 0177.51204
[37] Veech, W., Topological dynamics, Bull. Amer. Math. Soc., 83, 5, 775-830, (1977) · Zbl 0384.28018
[38] Weiss, B., A survey of generic dynamics, (Descriptive Set Theory and Dynamical Systems, Marseille-Luminy, 1996, (2000)), 273-291 · Zbl 0962.37001
[39] Ziegler, T., Universal characteristic factors and furstenberg averages, J. Amer. Math. Soc., 20, 1, 53-97, (2007), (electronic) · Zbl 1198.37014
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