In this paper we give a smooth linearization theorem for nonautonomous difference equations with a nonuniform strong exponential dichotomy. The linear part of such a nonautonomous difference equation is defined by a sequence of invertible linear operators on R d . Reducing the linear part to a bounded linear operator on a Banach space, we discuss the spectrum and its spectral gaps. Then we obtain a gap condition for C 1 linearization of such a nonautonomous difference equation. .Our theorems improve known results even in the case of uniform strong exponential dichotomies.

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C 1 linearization is of special interests because it can distinguish characteristic directions of dynamical systems. It is known that planar C 1,α contractions with a fixed point at the origin O admit C 1,β linearization with sufficiently small β > 0 if α = 1 and admit C 1,α linearization if (log |λ 1 |/ log |λ 2 |) − 1 < α 1, where λ 1 and λ 2 are eigenvalues of the linear parts of the contractions at O with 0 < |λ 1 | |λ 2 | < 1. In this paper we improve the lower bound of α to lower the condition of C 1 linearization for planar contractions. Furthermore, we prove that the derivatives of transformations in our C 1 linearization are Hölder continuous and give estimates for the Hölder exponent. Finally, we give a counter example to show that those estimates cannot be improved anymore.

In this paper we give a smooth linearization theorem for nonautonomous differential equations with a nonuniform strong exponential dichotomy. In terms of a discretized evolution operator with hyperbolic fixed point 0, we formulate its spectrum and then give a spectral bound condition for the linearization of such equations to be simultaneously differentiable at 0 and Hölder continuous near 0. Restricted to the autonomous case, our result is the first one that gives a rigorous proof for simultaneously differentiable and Hölder linearization of hyperbolic systems without any nonresonant conditions.