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

Volume 202 / 2011

Piotr Oprocha, Guohua Zhang Studia Mathematica 202 (2011), 261-288 MSC: Primary 37B05; Secondary 37B40, 37E05, 37E25. DOI: 10.4064/sm202-3-4

Abstract

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.

Authors

  • Piotr OprochaDepartamento 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
    e-mail
  • Guohua ZhangSchool 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
    e-mail

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