Nonautonomous Dynamics with Discrete Time and Topological Equivalence

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For evolution families with discrete time, we show that any exponential dichotomy is topologically equivalent to a certain normal form, in which the exponential behavior along the stable and unstable directions are multiples of the identity. We consider the general case of a generalized exponential dichotomy in which the usual exponential behavior is replaced by an arbitrary growth rate. In addition, we show that the topological equivalence between two evolution families with generalized exponential dichotomies can be completely characterized in terms of a notion of equivalence between the growth rates.