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A Grobman–Hartman theorem for nonuniformly hyperbolic dynamics. (English) Zbl 1099.37022

Summary: 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.

MSC:

37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37B55 Topological dynamics of nonautonomous systems
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
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