Nonuniform dichotomy spectrum and reducibility for nonautonomous equations

TL;DR

This paper introduces a nonuniform dichotomy spectrum for nonautonomous linear differential equations, generalizing existing spectral concepts, and demonstrates its utility through spectral and reducibility theorems.

Contribution

It defines a new spectral concept for nonautonomous equations with nonuniform hyperbolicity and proves key theorems establishing its properties and implications.

Findings

Defined nonuniform dichotomy spectrum for nonautonomous equations

Proved a spectral theorem for the new spectrum

Established a reducibility result based on the spectrum

Abstract

For nonautonomous linear differential equations with nonuniform hyperbolicity, we introduce a definition for nonuniform dichotomy spectrum, which can be seen as a generalization of Sacker-Sell spectrum. We prove a spectral theorem and use the spectral theorem to prove a reducibility result.