A Grobman–Hartman theorem for nonuniformly hyperbolic dynamics

Abstract

We establish a version of the Grobman-Hartman theorem in Banach spaces for nonuniformly hyperbolic dynamics. We also consider the case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. More precisely, we consider sequences of Lipschitz maps A m +f m such that the linear parts A m admit a nonuniform exponential dichotomy, and we establish the existence of a unique sequence of topological conjugacies between the maps A m + f m and A m . Furthermore, we show that the conjugacies are Hölder continuous, with Hölder exponent determined by the ratios of Lyapunov exponents with the same sign. To the best of our knowledge this statement appeared nowhere before in the published literature, even in the particular case of uniform exponential dichotomies, although some experts claim that it is well known in this case. We are also interested in the dependence of the conjugacies on the perturbations f m : we show that it is Hölder continuous, with the same Hölder exponent as the one for the conjugacies. We emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial. In particular, we need to consider several additional Lyapunov norms.

Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps

“…Unfortunately, there is no analog of Moser's theorem for nonlinear, nonautonomous area preserving two dimensional maps. Nevertheless, we will still use a "change of variables", or "conjugation" result that is a nonautonomous map version of the Hartman-Grobman theorem due to Barreira and Valls (2006).…”

“…There are two issues that we must immediately face in order for this approach to work as it did for the linear and nonlinear autonomous maps. One is the generalization of the Hartman-Grobman theorem to the setting on nonautonomous maps (this is dealt with in Barreira and Valls (2006)) and the other is the smoothness of the conjugation ("change of coordinates") since a derivative is required in the application of the chain rule (see (28)). …”

“…Precise assumptions on A n and f n (v) are given in Barreira and Valls (2006)). In particular A n is a hyperbolic operator, which for us is:…”

Smooth center manifolds for nonuniformly partially hyperbolic trajectories

“…This is certainly the case when (2) admits an exponential dichotomy: by an appropriate version of the Grobman-Hartman theorem, and under certain "smallness" assumptions on the perturbation, locally the two dynamics are topologically conjugate (we refer to [3] for detailed references; we note that [3] also considers the more general case of nonuniform exponential dichotomies). When Eq.…”

Reversibility of Conjugacies for Nonautonomous Equations

“…We use the notion of nonautonomous reversible equation discussed in [3]. Our approach is based in work in [1,2], where we obtain versions of the Grobman-Hartman theorem in the general setting of nonuniformly hyperbolic dynamics for nonautonomous differential equations in Banach spaces, respectively for discrete time and continuous time. In particular, it is shown in these papers that all conjugacies are Hölder continuous.…”