For any sufficiently small perturbation of a tempered exponential dichotomy, we obtain appropriate versions of the Grobman–Hartman theorem and of the Sternberg theorems both for finite and infinite regularity, thus providing, respectively, topological and smooth conjugacies. The constructions are heavily based on the existence of normal forms, with the resonances expressed in terms of the connected components of the nonuniform spectrum, which is a tempered version of the Sacker–Sell spectrum. In order to obtain specific tempered bounds for the derivatives up to a certain order, we first construct smooth stable and unstable invariant manifolds for any sufficiently small perturbation of a tempered exponential dichotomy. We also make several preparations of the dynamics so that the linear part has a block form, the nonlinear part has no terms up to a given order, and the stable and unstable manifolds coincide, respectively, with the stable and unstable spaces. The conjugacies are then constructed via the homotopy method.

中文翻译：

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非均匀双曲线下的平滑线性化

对于缓和指数二分法的任何足够小的扰动，我们获得了有限和无限正则性的 Grobman-Hartman 定理和 Sternberg 定理的适当版本，从而分别提供了拓扑共轭和平滑共轭。这些构造在很大程度上基于正常形式的存在，共振以非均匀谱的连接分量表示，这是 Sacker-Sell 谱的缓和版本。为了获得特定阶导数的特定缓和边界，我们首先为缓和指数二分法的任何足够小的扰动构造平滑的稳定和不稳定的不变流形。我们还做了一些动力学的准备，使线性部分具有块状，非线性部分没有达到给定阶数的项，稳定和不稳定流形分别与稳定和不稳定空间重合。然后通过同伦方法构建共轭。