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On weak mixing, minimality and weak disjointness of all iterates

Published online by Cambridge University Press:  14 October 2011

Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland (email:
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:


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.

Research Article
Copyright © Cambridge University Press 2011

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