A Franks' lemma that preserves invariant manifolds

23 pagesA 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