A Franks' lemma that preserves invariant manifolds (Q2813945)

The well-known Franks' lemma asserts that one can obtain 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. This lemma provides a useful tool in the study of perturbation theory in smooth dynamical systems. However, one does not control where the invariant manifolds of the orbit are, after perturbation.NEWLINENEWLINEIn the paper under review, it is shown that if the perturbed 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 provides a more general tool in \(C^1\) dynamics that has many consequences. Moreover, there are some applications of the main results, such as a generic dichotomy between dominated splittings and small stable/unstable angles inside homoclinic classes.