We prove that the conjugacies in the Grobman-Hartman theorem are always Holder continuous, with Holder exponent determined by the ratios of Lyapunov exponents with the same sign. We also consider the case of hyperbolic trajectories of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. All the results are obtained in Banach spaces. It is common knowledge that some authors claimed that the Holder regularity of the conjugacies is well known, however without providing any reference. In fact, to the best of our knowledge, the proof appears nowhere in the published literature.
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