  A linear system of difference equations and a nonlinear perturbation are
considered.
  We propose sufficient conditions to ensure that the homeomorphism of
topological equivalence between them is actually a $C^1$ diffeomorphism. These
conditions consider that the linear part satisfies a dichotomy on the positive
half-line, while the perturbation satisfies boundness and Lipschitzness
conditions, and anothers that are tailored for our goal. The family of
dichotomies that we study consider (but are not limited to) the exponential and
nonuniform exponential dichotomies. We discuss the possibility of extending
this result to a higher class of differentiability.
