$f$-Frequently hypercyclic $C_{0}$-semigroups on complex sectors

TL;DR

This paper studies various types of hypercyclicity of $C_0$-semigroups on complex sectors, providing structural results, analyzing specific classes like translation semigroups, and illustrating findings with examples.

Contribution

It introduces new structural results for ${ m extbf{F}}$-frequently hypercyclic $C_0$-semigroups and explores generalized hypercyclicity in weighted function spaces.

Findings

Structural results for ${ m extbf{F}}$-frequently hypercyclic semigroups

Analysis of generalized hypercyclic translation semigroups

Examples illustrating hypercyclic behavior on complex sectors

Abstract

We analyze $f$-frequently hypercyclic, $q$-frequently hypercyclic ($q> 1$) and frequently hypercyclic $C_{0}$-semigroups ($q=1$) defined on complex sectors, working in the setting of separable infinite-dimensional Fr\'echet spaces. Some structural results of ours are given for a general class of ${\mathcal F}$-frequently hypercyclic $C_{0}$-semigroups, as well. We investigate generalized frequently hypercyclic translation semigroups and generalized frequently hypercyclic semigroups induced by semiflows on weighted function spaces. Several illustrative examples are presented.