Discrete $\mu$-dichotomy spectrum: beyond uniformity and new insights

TL;DR

This paper advances the spectral theory of nonautonomous linear difference systems by introducing a new nonuniform $rac{ ext{dichotomy spectrum}},$ exploring unconventional behaviors, and proposing a conjecture to unify spectral results beyond traditional assumptions.

Contribution

It introduces a new conceptualization of the nonuniform $rac{ ext{dichotomy spectrum}}$ and proposes a conjecture to derive spectral theorems in this broader setting.

Findings

Nonuniform exponential dichotomy spectrum is not preserved under weak kinematic similarity.

New properties of nonautonomous systems are identified and characterized.

A new spectrum between traditional and slow nonuniform $rac{ ext{dichotomy spectrum}}$ is proposed.

Abstract

We develop spectral theorems for nonautonomous linear difference systems, considering different types of $\mu$-dichotomies, both uniform and nonuniform. In the nonuniform case, intriguing scenarios emerge -- that have been employed but whose consequences have not been thoroughly explored -- which surprisingly exhibit unconventional behavior. These particular cases motivate us to introduce two novel properties of nonautonomous systems (even in the continuous-time framework), which appear to have been overlooked in the existing literature. Additionally, we introduce a new conceptualization of a nonuniform $\mu$-dichotomy spectrum, which lies between the traditional nonuniform $\mu$-dichotomy spectrum and the slow nonuniform $\mu$-dichotomy spectrum. Moreover, and this is particularly noteworthy, we propose a conjecture that enables the derivation of spectral theorems in this new setting. Finally, contrary to what has been believed in recent years, through the lens of optimal ratio maps, we show that the nonuniform exponential dichotomy spectrum is not preserved between systems that are weakly kinematically similar.