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Recurrence properties of hypercyclic operators (1410.1349v1)

Published 6 Oct 2014 in math.FA and math.DS

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

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