Nonautonomous Normal Forms with Parameters

For a nonautonomous dynamics with discrete time depending on a parameter, we construct normal forms that have the same regularity as the original dynamics. A principal difficulty is that the resonances may depend on the parameter. The proof consists of three main elements that are interesting in their own right. First, we show that the spectrum of a nonautonomous linear dynamics does not vary much under perturbations. Second, we use this perturbation result to reduce the linear dynamics to a block-diagonal form with respect to a splitting that is independent of the parameter. Finally, we show that the normal forms are as regular as the original dynamics provided that the resonance conditions are relaxed to allow a certain spectral gap.