Lipschitz shadowing for contracting/expanding dynamics on average

Abstract

We prove that Lipschitz perturbations of nonautonomous contracting or expanding linear dynamics are Lipschitz shadowable provided that the Lipschitz constants are small on average. This is in sharp contrast with previous results where the Lipschitz constants are assumed to be uniformly small. Moreover, we show by means of an example that a natural extension of these results to the context of linear dynamics admitting an exponential dichotomy does not hold in general.