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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:
 37B05 Transformations and group actions with special properties 37B40 Topological entropy 37E05 Maps of the interval (piecewise continuous, continuous, smooth) 37E25 Maps of trees and graphs
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