A well-known lemma by John Franks asserts that one can realise any
perturbation of the derivative of a diffeomorphism $f$ along a periodic orbit
by a $C^1$-perturbation of the whole diffeomorphism on a small neighbourhood of
the orbit. However, it does not provide any information on the behaviour of the
invariant manifolds of the orbit after perturbation.

In this paper we show that if the perturbated derivative can be joined from
the initial derivative by a path, and if some strong stable or unstable
directions of some indices exist along that path, then the corresponding
invariant manifolds can be preserved outside of a small neighbourhood of the
orbit. We deduce perturbative results on homoclinic classes, in particular a
generic dichotomy between dominated splitting and small stable/unstable angles
inside homoclinic classes.