Expansivity and Shadowing in Linear Dynamics

TL;DR

This paper explores the relationships between expansivity, shadowing, and other dynamical properties of linear operators on Banach spaces, providing characterizations and new examples that highlight the complexity of infinite-dimensional linear dynamics.

Contribution

It establishes new links between expansivity, hypercyclicity, and spectrum, and constructs nonhyperbolic invertible operators with shadowing, advancing understanding in linear dynamics.

Findings

Complete characterizations of weighted shifts on c0 and ℓp spaces.

Existence of nonhyperbolic invertible operators with shadowing property.

New relationships between expansivity and spectral properties.

Abstract

In the early 1970's Eisenberg and Hedlund investigated relationships between expansivity and spectrum of operators on Banach spaces. In this paper we establish relationships between notions of expansivity and hypercyclicity, supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space is $c_0$ or $\ell_p$ ($1 \leq p < \infty$), we give complete characterizations of weighted shifts which satisfy various notions of expansivity. We also establish new relationships between notions of expansivity and spectrum. Moreover, we study various notions of shadowing for operators on Banach spaces. In particular, we solve a basic problem in linear dynamics by proving the existence of nonhyperbolic invertible operators with the shadowing property. This also contrasts with the expected results for nonlinear dynamics on compact manifolds, illuminating the richness of dynamics of infinite dimensional linear operators.