Smooth Linearization of Nonautonomous Difference Equations with a Nonuniform Dichotomy

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.