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On weak mixing, minimality and weak disjointness of all iterates

Published online by Cambridge University Press:  14 October 2011

DOMINIK KWIETNIAK
Affiliation:
Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland (email: dominik.kwietniak@uj.edu.pl)
PIOTR OPROCHA
Affiliation:
AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Kraków, Poland Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland (email: oprocha@agh.edu.pl)

Abstract

This article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map f×f2×⋯×fm, where f:XX is a topological dynamical system on a compact metric space. The theorem stating that a weakly mixing and strongly transitive system is Δ-transitive is extended to a non-invertible case with a simple proof. Two examples are constructed, answering the questions posed by Moothathu [Diagonal points having dense orbit. Colloq. Math. 120(1) (2010), 127–138]. The first one is a multi-transitive non-weakly mixing system, and the second one is a weakly mixing non-multi-transitive system. The examples are special spacing shifts. The latter shows that the assumption of minimality in the multiple recurrence theorem cannot be replaced by weak mixing.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2011

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