Recurrence properties of hypercyclic operators

TL;DR

This paper introduces a new concept called $ ext{A}$-frequent hypercyclicity, generalizing existing notions, and provides a criterion to characterize it, especially for weighted shift operators, while exploring properties of hypercyclic sets.

Contribution

It proposes the $ ext{A}$-Frequent Hypercyclicity Criterion, extending hypercyclicity theory, and analyzes the properties of hypercyclic sets and reiteratively hypercyclic operators.

Findings

The $ ext{A}$-Frequent Hypercyclicity Criterion characterizes $ ext{A}$-frequent hypercyclicity for weighted shifts.

The paper identifies properties of density for hypercyclic sets.

It introduces and studies reiteratively 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.