Recurrence properties of hypercyclic operators

Abstract

We generalize the notions of hypercyclic operators, $\mathfrak{U}$-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new notion of hypercyclicity, called $\mathcal{A}$-frequent hypercyclicity. We then state an $\mathcal{A}$-Frequent Hypercyclicity Criterion, inspired from the Hypercyclicity Criterion and the Frequent Hypercyclicity Criterion, and we show that this criterion characterizes the $\mathcal{A}$-frequent hypercyclicity for weighted shifts. We finish by investigating which kind of properties of density can have the sets $N(x,U)=\{n\in \mathbb{N}:T^{n}x\in U\}$ for a given hypercyclic operator and study the new notion of reiteratively hypercyclic operators.

24 pages

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