Exponential Dichotomy for Noninvertible Linear Difference Equations

TL;DR

This paper investigates exponential dichotomies for noninvertible linear difference equations, characterizing projection uniqueness, extension conditions, and stability under perturbations, extending existing theories to noninvertible cases.

Contribution

It provides a complete characterization of projections in exponential dichotomies for noninvertible difference equations and extends stability results to multiplicative perturbations.

Findings

Projection uniqueness depends on the domain (Z, Z+, Z-)

Conditions for extending dichotomies without changing projections

Roughness theorem applies to multiplicative perturbations

Abstract

In this article we study exponential dichotomies for noninvertible linear difference equations in finite dimensions. After giving the definition, we study the extent to which the projection $P(k)$ in a dichotomy is unique. For equations on $\mathbb{Z}$ it is unique but for equations on $\mathbb{Z}_+$ only its range is unique and for $\mathbb{Z}_-$ only its nullspace. Here we strengthen Kalkbrenner's results and give a complete characterization of all possible projections. Next we study the possibility of extending the dichotomy to a larger interval. We reproduce the results of P\" otzsche but also show exactly when the original projection remains unchanged. Next we prove that the roughness theorem, well known for additive perturbations, holds for multiplicative perturbations also. The proof uses ideas of Zhou, Lu and Zhang. Finally, following Ducrot, Magal and Seydi, we mention that the results by Palmer on finite time conditions on dichotomy for the invertible case can be extended to the noninvertible case.