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Linear dynamics and recurrence properties defined via essential idempotents of $β\mathbb{N}$

Consider $\mathscr{F}$ a non-empty set of subsets of $\mathbb{N}$. An operator $T$ on $X$ satisfies property $\mathcal{P}_{\mathscr{F}}$ if for any $U$ non-empty open set in $X$, there exists $x\in X$ such that $\{n\in\mathbb{N}: T^nx\in U\}\in \mathscr{F}$. Let $\overline{\mathcal{BD}}$ the collection of sets in $\mathbb{N}$ with positive upper Banach density. Our main result is a characterization of sequence of operators satisfying property $\mathcal{P}_{\overline{\mathcal{BD}}}$, for which we have used a strong result of Bergelson and Mccutcheon in the vein of Szemerédi's theorem. It turns out that operators having property $\mathcal{P}_{\overline{\mathcal{BD}}}$ satisfy a kind of recurrence described in terms of essential idempotents of $β\mathbb{N}$. We will also discuss the case of weighted backward shifts. Finally, we obtain a characterization of reiteratively hypercyclic operators.