Smooth linearization of nonautonomous dynamics under polynomial behaviour

TL;DR

This paper establishes new conditions for smoothly linearizing nonautonomous systems with polynomial dichotomies, expanding beyond the traditional exponential dichotomy assumptions by leveraging a reparametrization of time.

Contribution

It introduces novel criteria for smooth linearization under polynomial dichotomies, contrasting with prior exponential-based conditions, and utilizes a reparametrization technique to connect polynomial and exponential dichotomies.

Findings

Derived conditions for smooth linearization under polynomial dichotomies

Established spectral gap and band conditions for the polynomial dichotomy spectrum

Linked polynomial and exponential dichotomies through time reparametrization

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 nonuniform polynomial dichotomy and that the associated polynomial dichotomy spectrum exhibits appropriate spectral gap and spectral band conditions. This 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 polynomial and exponential dichotomies via a suitable reparametrization of time.