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We prove that the conjugacies in the Grobman-Hartman theorem are
always Hölder continuous, with Hölder 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 Hölder 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.