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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm202-3-4,
author = {Piotr Oprocha and Guohua Zhang},
title = {On local aspects of topological weak mixing in dimension one and beyond},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {261-288},
zbl = {1217.37012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm202-3-4}
}
Piotr Oprocha; Guohua Zhang. On local aspects of topological weak mixing in dimension one and beyond. Studia Mathematica, Tome 204 (2011) pp. 261-288. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm202-3-4/