Optimal regularity of robustness for parameterized perturbations

For exponential dichotomies defined by nonautonomous linear equations, we show that sufficiently small C1-parameterized perturbations originate a family of exponential dichotomies of class C1 in the parameter. We consider the general case of nonuniform exponential dichotomies, and also the general case of arbitrary growth rates of the form eλρ(t) where ρ is an arbitrary function. This includes the usual exponential behavior as a very special case when ρ(t)=t.