Linear dynamics and recurrence properties defined via essential idempotents of βn

Consider , a non-empty set of subsets of . An operator on satisfies property if, for any non-empty open set in , there exists such that . Let be the collection of sets in with positive upper Banach density. Our main result is a characterization of a sequence of operators satisfying property , for which we have used a deep result of Bergelson and McCutcheon in the vein of Szemerédi's theorem. It turns out that operators having property satisfy a kind of recurrence described in terms of essential idempotents of . We will also discuss the case of weighted backward shifts. Finally, we obtain a characterization of reiteratively hypercyclic operators.