Chain Recurrence and Positive Shadowing in Linear Dynamics

TL;DR

This paper investigates the properties of chain recurrence and shadowing in linear dynamical systems, revealing that chain recurrence simplifies to a unique invariant subspace and linking shadowing to frequent hypercyclicity.

Contribution

It establishes that linear operators have a single chain recurrent set which is a closed invariant subspace, and connects shadowing with frequent hypercyclicity in linear dynamics.

Findings

Linear operators have a unique chain recurrent set that is a closed invariant subspace.

Chain transitivity with shadowing implies frequent hypercyclicity.

Shadowing hypercyclic systems are also frequently hypercyclic.

Abstract

We study shadowing and chain recurrence in the context of linear operators acting on Banach spaces or even on normed vector spaces. We show that for linear operators there is only one chain recurrent set, and this set is actually a closed invariant subspace. We prove that every chain transitive linear dynamical system with the shadowing property is frequently hypercyclic, and as a corollary obtain that every shadowing hypercyclic linear dynamical system is frequently hypercyclic.