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On recurrence over subsets and weak mixing. (English) Zbl 1352.37032

Summary: We study properties of weakly mixing sets (of order \(n\)) in relation to proximality, sensitivity, scrambled tuples, Xiong chaotic sets and independent sets. Our main emphasis is on the structure of the set of transfer times \(N(U\cap A,V)\) between open sets \(U\) and \(V\), both intersecting a weakly mixing set \(A\). We find several conditions on properties of the set \(A\) that are equivalent to weak mixing. We also prove that on topological graphs weakly mixing sets of order \(2\) can be approximated arbitrarily closely by a weakly mixing set of all orders. This property is known to hold on the unit interval but is not true in general (there are systems with weakly mixing sets of order \(n\) but not \(n+1\)).

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B40 Topological entropy
37E05 Dynamical systems involving maps of the interval
37E25 Dynamical systems involving maps of trees and graphs
37A25 Ergodicity, mixing, rates of mixing
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