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

Piotr Oprocha ¹   ; Guohua Zhang ²

¹ Departamento de Matemáticas Universidad de Murcia Campus de Espinardo 30100 Murcia, Spain and Faculty of Applied Mathematics AGH University of Science and Technology al. Mickiewicza 30 30-059 Kraków, Poland
² School of Mathematical Sciences and LMNS Fudan University Shanghai 200433, China and School of Mathematics and Statistics University of New South Wales Sydney, NSW 2052, Australia

Studia Mathematica, Tome 202 (2011) no. 3, pp. 261-288

Résumé

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.

Détail

DOI : 10.4064/sm202-3-4

Keywords: introduce concept weakly mixing sets order contrast weak mixing maps weakly mixing set order does have weakly mixing order strictly speaking construct minimal invertible dynamical system which contains non trivial weakly mixing set order whereas does contain non trivial weakly mixing set order dimension difference much visible since prove every continuous map topological graph itself has positive topological entropy only contains non trivial weakly mixing set order only contains non trivial weakly mixing set orders