Optimal growth of upper frequently hypercyclic functions for some weighted Taylor shifts

TL;DR

This paper investigates the optimal growth rates of hypercyclic and frequently hypercyclic functions under weighted Taylor shifts on the unit disc, unifying various notions of hypercyclicity.

Contribution

It introduces a unified framework for analyzing growth rates of hypercyclic functions across different levels of frequent hypercyclicity for weighted Taylor shifts.

Findings

Established optimal growth bounds for hypercyclic functions.

Unified intermediate notions of upper frequent hypercyclicity.

Extended previous results to broader classes of hypercyclicity.

Abstract

We are interested in the optimal growth in terms of $L^p$-averages of hypercyclic and $\mathcal{U}$-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic function on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between the $\mathcal{U}$-frequent hypercyclicity and the hypercyclicity.