2009/12/06 by Nikolaz Gourmelon, Gourmelon, Nikolaz · 1 citation
Mathematics · #37C20 #37C25 #37C29 #37D10 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37C20 #msc:37C25 #msc:37C29 #msc:37D10
paper · pdf · doi:10.48550/arxiv.0912.1121
36 pages, 4 figures
arxiv created 2014/09/27 · arxiv updated 2014/09/30
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 C1-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 C1-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