On weak mixing, minimality and weak disjointness of all iterates

This article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map f x f(2) x . . . x f(m), where f : X -> 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 [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.