A quantitative interpretation of the frequent hypercyclicity criterion

TL;DR

This paper provides a quantitative analysis of the Frequent Hypercyclicity Criterion, showing that operators satisfying it are A-frequently hypercyclic with respect to certain weighted densities, and presents an example distinguishing different frequency scales.

Contribution

It introduces a refined framework for measuring the frequency of hypercyclicity using weighted densities and constructs an example illustrating the distinctions between these scales.

Findings

Operators satisfying the criterion are A-frequently hypercyclic for certain weighted densities.

Different weighted density scales can quantify the frequency of hypercyclicity.

An example of a weighted shift is provided that is frequently hypercyclic but not A-frequently hypercyclic on a specific scale.

Abstract

We give a quantitative interpretation of the Frequent Hypercyclicity Criterion. Actually we show that an operator which satisfies the Frequent Hypercyclicity Criterion is necessarily A-frequently hypercyclic, where A refers to some weighted densities sharper than the natural lower density. In that order, we exhibit different scales of weighted densities that are of interest to quantify the "frequency" measured by the Frequent Hypercyclicity Criterion. Moreover we construct an example of unilateral weighted shift which is frequently hypercyclic but not A-frequently hypercyclic on a particular scale.