An Isotopic Perturbation Lemma Along Periodic Orbits

Nicolas Gourmelon (1, 2, 3)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 IMB - Institut de Mathématiques de Bordeaux
3 IMPA - Instituto Nacional de Matemática Pura e Aplicada

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.

Mots clés: Franks Lemma; periodic point; saddle point; linear cocycle; perturbation; stable/unstable manifold; dominated splitting; homoclinic tangency; small angles

Domaines: Systèmes dynamiques [math.DS]

Nicolas Gourmelon. An Isotopic Perturbation Lemma Along Periodic Orbits. 2012. ⟨hal-00769205⟩