The interplay of $\mu$-dichotomy, bounded growth, and spectral properties via growth rate comparisons

TL;DR

This paper explores how different growth rates affect the dichotomy spectrum of nonautonomous linear systems, establishing comparison criteria and classifying spectra through equivalence relations in both discrete and continuous time.

Contribution

It introduces new comparison criteria for growth rates and their impact on the dichotomy spectrum, along with a classification framework based on equivalence relations.

Findings

Faster growth rates compress the dichotomy spectrum.

Slower growth rates expand the spectrum.

Existence of at most one suitable growth rate for bounded growth and dichotomy properties.

Abstract

We investigate the behavior of the dichotomy spectrum of nonautonomous linear systems under general growth rates. By introducing comparison criteria we clarify how $\mu$-dichotomy and $\mu$-bounded growth interact. We also study the evolution of the dichotomy spectrum under these comparisons, revealing that faster growth rates compress the spectrum, while slower growth rates expand it. Moreover, we introduce equivalence relations on the set of growth rates, which enable us to establish that, for any given system, there exists -- up to equivalence -- at most one growth rate under which both properties, bounded growth and dichotomy, hold. Finally, we show that these equivalence relations lead to a classification of dichotomy spectra. Our results are valid in both discrete and continuous time settings.