A geometrical proof of the persistence of normally hyperbolic submanifolds

Pierre Berger (1) , Abed Bounemoura (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 IMPA - Instituto Nacional de Matemática Pura e Aplicada

We present a simple, computation free and geometrical proof of the following classical result: for a diffeomorphism of a manifold, any compact submanifold which is invariant and normally hyperbolic persists under small perturbations of the diffeomorphism. The persistence of a Lipschitz invariant submanifold follows from an application of the Schauder fixed point theorem to a graph transform, while smoothness and uniqueness of the invariant submanifold are obtained through geometrical arguments. Moreover, our proof provides a new result on persistence and regularity of ''topologically" normally hyperbolic submanifolds, but without any uniqueness statement.

Domaines: Systèmes dynamiques [math.DS]

Pierre Berger, Abed Bounemoura. A geometrical proof of the persistence of normally hyperbolic submanifolds. Dynamical Systems, 2013, 28 (4), pp.567-581. ⟨hal-00623713⟩