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Shadowing for infinite dimensional dynamics and exponential trichotomies

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.

Shadowing, Nonautonomus systems, Exponential trichotomies, Nonlinear perturbations, Hyers-Ulam stability

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