Shadowing for nonautonomous dynamics

Abstract

We prove that whenever a sequence of invertible and bounded operators $(A_m{)}_{m\in \mathbb{Z}}$ acting on a Banach space $X$ admits an exponential dichotomy and a sequence of differentiable maps $f_m:X\to X$, $m\in \mathbb{Z}$, has bounded and Hölder derivatives, the nonautonomous dynamics given by $x_{m+1}=A_m{x}_m+f_m(x_m)$, $m\in \mathbb{Z}$ has various shadowing properties. Hence, we extend recent results of Bernardes Jr. et al. in several directions. As a nontrivial application of our results, we give a new proof of the nonautonomous Grobman-Hartman theorem.