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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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