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On local aspects of topological weak mixing in dimension one and beyond. (English) Zbl 1217.37012

Summary: We introduce the concept of weakly mixing sets of order n and show that, in contrast to weak mixing of maps, a weakly mixing set of order n does not have to be weakly mixing of order n+1. Strictly speaking, we construct a minimal invertible dynamical system which contains a non-trivial weakly mixing set of order 2, whereas it does not contain any non-trivial weakly mixing set of order 3.

In dimension one this difference is not that much visible, since we prove that every continuous map f from a topological graph into itself has positive topological entropy if and only if it contains a non-trivial weakly mixing set of order 2 if and only if it contains a non-trivial weakly mixing set of all orders.

MSC:
37B05Transformations and group actions with special properties
37B40Topological entropy
37E05Maps of the interval (piecewise continuous, continuous, smooth)
37E25Maps of trees and graphs