2012/12/29 by Nicolas Gourmelon, Gourmelon, Nicolas · 2 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS
paper · pdf · doi:10.48550/arxiv.1212.6638
51 pages. arXiv admin note: substantial text overlap with arXiv:0912.1121
arxiv created 2012/12/29 · openalex publication_date 2012/12/29 · arxiv updated 2014/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.