Smooth Linearization of Nonautonomous Difference Equations with a Nonuniform Dichotomy

TL;DR

This paper establishes a smooth linearization theorem for nonautonomous difference equations with nonuniform strong exponential dichotomies, extending previous results and including infinite-dimensional cases.

Contribution

It provides a new gap condition for $C^1$ linearization of nonautonomous difference equations with nonuniform dichotomies, improving upon existing results.

Findings

Derived a gap condition for $C^1$ linearization.

Extended the linearization results to infinite-dimensional spaces.

Improved upon known results even for uniform strong exponential dichotomies.

Abstract

In this paper we give a smooth linearization theorem for nonautonomous difference equations with a nonuniform strong exponential dichotomy. The linear part of such a nonautonomous difference equation is defined by a sequence of invertible linear operators on $\mathbb{R}^d$. Reducing the linear part to a bounded linear operator on a Banach space, we discuss the spectrum and its spectral gaps. Then we obtain a gap condition for $C^1$ linearization of such a nonautonomous difference equation. We finally extend the result to the infinite dimensional case. Our theorems improve known results even in the case of uniform strong exponential dichotomies.