On local aspects of topological weak mixing in dimension one and beyond.

*(English)*Zbl 1217.37012Summary: 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.