On weak mixing, minimality and weak disjointness of all iterates

Details On weak mixing, minimality and weak disjointness of all iterates On weak mixing, minimality and weak disjointness of all iterates

2010-09-13

arxiv.org/abs/1009.2277v4

Mathematics

Dynamical Systems

10 pages, to appear in Ergodic Theory and Dynamical Systems

Scientific paper

10.1017/S0143385711000599

The article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map $f\times f^2 \times...\times f^m$, where $f\colon X\ra X$ is a topological dynamical system on a compact metric space. The theorem stating that a weakly mixing and strongly transitive system is $\Delta$-transitive is extended to a non-invertible case with a simple proof. Two examples are constructed, answering the questions posed by Moothathu [Colloq. Math. 120 (2010), no. 1, 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 later shows that the assumption of minimality in the Multiple Recurrence Theorem can not be replaced by weak mixing.

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