# Shadowing for infinite dimensional dynamics and exponential trichotomies

## TL;DR

This paper proves shadowing properties for nonautonomous Banach space dynamics with exponential trichotomies under small Lipschitz perturbations, and explores stability and conjugacy results.

## Contribution

It establishes shadowing and stability results for infinite-dimensional nonautonomous systems with exponential trichotomies, including converse theorems in finite dimensions.

## Key findings

- Shadowing holds for small Lipschitz perturbations in Banach spaces.
- Converse shadowing results are proved in finite-dimensional invertible cases.
- Applications include Hyers-Ulam stability and a generalized Grobman-Hartman theorem.

## Abstract

Let $(A_m)_{m\in \Z}$ be a sequence of bounded linear maps acting on an arbitrary Banach space $X$ and admitting an exponential trichotomy and let $f_m:X\to X$ be a Lispchitz map for every $m\in \Z$. We prove that whenever the Lipschitz constants of $f_m$, $m\in \Z$, are uniformly small, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in \Z$, has various types of shadowing. Moreover, if $X$ is finite dimensional and each $A_m$ is invertible we prove that a converse result is also true. Furthermore, we get similar results for one-sided and continuous time dynamics. As applications of our results we study the Hyers-Ulam stability for certain difference equations and we obtain a very general version of the Grobman-Hartman's theorem for nonautonomous dynamics.