Shadowing and structural stability in linear dynamical systems

TL;DR

This paper explores the relationship between hyperbolicity, shadowing, and structural stability in linear dynamical systems, providing new examples and characterizations of operators with these properties.

Contribution

It demonstrates that not all structurally stable operators are hyperbolic and characterizes operators with shadowing and stability properties, especially weighted shifts.

Findings

Existence of structurally stable operators that are not hyperbolic.

Hyperbolic operators are exactly those that are expansive and have shadowing.

Characterization of weighted shifts on certain spaces satisfying shadowing.

Abstract

A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by P. Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by J. Palis and C. Pugh around 1968. We will exhibit examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces $c_0(\mathbb{Z})$ and $\ell_p(\mathbb{Z})$ ($1 \leq p < \infty$) that satisfy the shadowing property.