  A well-known lemma by John Franks asserts that one obtains any perturbation
of the derivative of a diffeomorphism along a periodic orbit by a
$C^1$-perturbation of the whole diffeomorphism on a small neighbourhood of the
orbit. However, one does not control where the invariant manifolds of the orbit
are, after perturbation.
  We show that if the perturbated derivative is obtained by an isotopy along
which some strong stable/unstable manifolds of some dimensions exist, then the
Franks perturbation can be done preserving the corresponding stable/unstable
semi-local manifolds. This is a general perturbative tool in $C^1$-dynamics
that has many consequences. We give simple examples of such consequences, for
instance a generic dichotomy between dominated splitting and small
stable/unstable angles inside homoclinic classes

Dynamical Systems