Normal forms via nonuniform hyperbolicity

Abstract

Abstract We develop a normal form theory for a nonautonomous dynamics x m + 1 = A m x m + f m ( x m ) with discrete time, based on the nonuniform spectrum of the sequence of matrices A m . In particular, we show that any nonresonant terms of the perturbations f m can be eliminated through an appropriate coordinate change, with the resonances expressed in terms of the connected components of the nonuniform spectrum. The latter is defined in terms of the notion of a nonuniform exponential dichotomy with a small nonuniform part, which is ubiquitous in the context of ergodic theory. We first make a preparation of the linear part of the dynamics that is of independent interest: we show that any sequence of matrices with a bounded nonuniform spectrum can be reduced to a sequence of matrices in block form via a Lyapunov coordinate change. This allows maintaining the Lyapunov exponents as well as the nonuniform spectrum. As further developments, we describe normal form theories in two additional contexts: we consider nonuniformly hyperbolic cocycles over a diffeomorphism of a compact manifold as well as perturbations f m of a sequence of compact linear operators A m on a Banach space. The latter includes the particular case of a sequence of matrices that need not be invertible.