# $\mathcal{U}$-Frequent hypercyclicity notions and related weighted densities

## TL;DR

This paper explores intermediate dynamical properties between $$-frequent and reiterative hypercyclicity by analyzing weighted upper densities, revealing distinctions in chaos implications and product stability of these notions.

## Contribution

It introduces and investigates weighted upper densities between known hypercyclicity notions, clarifying their relationships and stability under product operations.

## Key findings

- Chaos does not imply $$-frequent hypercyclicity with any weighted density.
- Product of $$-frequently hypercyclic operators remains $$-frequently hypercyclic.
- Product of reiteratively hypercyclic operators remains reiteratively hypercyclic.

## Abstract

We study dynamical notions lying between $\mathcal{U}$-frequent hypercyclicity and reiterative hypercyclicity by investigating weighted upper densities between the unweighted upper density and the upper Banach density. While chaos implies reiterative hypercyclicity, we show that chaos does not imply $\mathcal{U}$-frequent hypercyclicity with respect to any weighted upper density. Moreover, we show that if $T$ is $\mathcal{U}$-frequently hypercyclic (resp. reiteratively hypercyclic) then the n-fold product of $T$ is still $\mathcal{U}$-frequently hypercyclic (resp. reiteratively hypercyclic) and that this implication is also satisfied for each of the considered $\mathcal{U}$-frequent hypercyclicity notions.