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A linear system of difference equations and a nonlinear perturbation areconsidered.  We propose sufficient conditions to ensure that the homeomorphism oftopological equivalence between them is actually a $C^1$ diffeomorphism. Theseconditions consider that the linear part satisfies a dichotomy on the positivehalf-line, while the perturbation satisfies boundness and Lipschitznessconditions, and anothers that are tailored for our goal. The family ofdichotomies that we study consider (but are not limited to) the exponential andnonuniform exponential dichotomies. We discuss the possibility of extendingthis result to a higher class of differentiability.

A note on the differentiability of Palmer's topological equivalence for discrete systems