Trichotomy for the orbits of a hypercyclic operator on a Banach space

TL;DR

This paper establishes a three-way classification of the behavior of orbits of hypercyclic operators on Banach spaces, revealing diverse asymptotic properties and providing concrete examples for each case.

Contribution

It introduces a trichotomy for the orbits of hypercyclic operators, detailing their mean asymptotic and divergence behaviors, with explicit examples demonstrating all cases.

Findings

All three cases of the trichotomy can occur in practice.

Different vectors exhibit distinct mean asymptotic behaviors.

Weighted backward shifts on ℓ^p exemplify each case.

Abstract

We obtain a trichotomy for the orbits of a hypercyclic operator $T$ on a separable Banach space $X$: (1) every vector is mean asymptotic to zero; (2) generic vectors are absolutely mean irregular; (3) every hypercyclic vector is mean divergent to infinity. Examples of weighted backward shifts on $\ell^p$ show that all three cases can happen.