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 \({N(x, U)=\{n\in \mathbb{N}:T^nx\in U\}}\) for a given hypercyclic operator and study the new notion of reiteratively hypercyclic operators.