Almost reducibility of linear difference systems from a spectral point of view

TL;DR

This paper demonstrates that under certain conditions, linear difference systems can be nearly simplified to diagonal form related to their spectral properties, with an example illustrating the concept of diagonally significant systems.

Contribution

It introduces conditions for almost reducibility of linear difference systems to diagonal form linked to the spectrum, and provides a key example of diagonally significant systems.

Findings

Systems are almost reducible to diagonal form within the spectrum.

An example of diagonally significant system is constructed.

The spectral approach is effective for analyzing reducibility.

Abstract

We prove that, under some conditions, a linear nonautonomous difference system is Bylov's almost reducible to a diagonal one whose terms are contained in the Sacker and Sell spectrum of the original system. We also provide an example of the concept of diagonally significant system, recently introduced by P\"otzche. This example plays an essential role in the demonstration of our results.