Shadowing for infinite dimensional dynamics and exponential trichotomies. Issue 3 (June 2021) - Backes, Lucas; Dragičević, Davor
Abstract : Let (A_m)_m 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 → X be a Lispchitz map for every mZ . We prove that whenever the Lipschitz constants of f_m, m Z , are uniformly small, the nonautonomous dynamics given by x_m+1=A_mx_m+f_m(x_m), mZ , 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.