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

Piotr Oprocha; Guohua Zhang

Studia Mathematica (2011)

  • Volume: 202, Issue: 3, page 261-288
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Piotr Oprocha, and Guohua Zhang. "On local aspects of topological weak mixing in dimension one and beyond." Studia Mathematica 202.3 (2011): 261-288. <http://eudml.org/doc/285754>.

@article{PiotrOprocha2011,
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. },
author = {Piotr Oprocha, Guohua Zhang},
journal = {Studia Mathematica},
keywords = {transitive set; weakly mixing set (of order 2); topological entropy},
language = {eng},
number = {3},
pages = {261-288},
title = {On local aspects of topological weak mixing in dimension one and beyond},
url = {http://eudml.org/doc/285754},
volume = {202},
year = {2011},
}

TY - JOUR
AU - Piotr Oprocha
AU - Guohua Zhang
TI - On local aspects of topological weak mixing in dimension one and beyond
JO - Studia Mathematica
PY - 2011
VL - 202
IS - 3
SP - 261
EP - 288
AB - 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.
LA - eng
KW - transitive set; weakly mixing set (of order 2); topological entropy
UR - http://eudml.org/doc/285754
ER -

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