Smooth linearization of nonautonomous dynamics under general dichotomic behaviour

TL;DR

This paper establishes new conditions for smooth linearization of nonautonomous systems using a general $ichotomy$, broadening classical results beyond exponential dichotomies.

Contribution

It introduces a general $ichotomy spectrum framework for linearization, encompassing exponential, polynomial, and logarithmic dichotomies, with new spectral gap conditions.

Findings

Provides conditions for smooth linearization under $ichotomy spectrum.

Shows $ichotomy includes classical exponential, polynomial, and logarithmic cases.

Uses reparametrization of time to relate $ichotomy$ to exponential dichotomy.

Abstract

The main purpose of this paper is to formulate new conditions for smooth linearization of nonautonomous systems with discrete and continuous time. Our results assume that the linear part admits a very general form of dichotomy known as $\mu$-dichotomy and that the associated $\mu$-dichotomy spectrum exhibits appropriate spectral gap and spectral band conditions. We observe that our notion of $\mu$-dichotomy encompasses the classical notions of exponential, polynomial and logarithmic dichotomies as very particular cases. In particular, our result is in sharp contrast to most of the previous results in the literature which assumed that the linear part admits an exponential dichotomy. Our techniques exploit the relationship between $\mu$-dichotomy and exponential dichotomy via a suitable reparametrization of time.