Article Dans Une Revue Dynamical Systems Année : 2013

A geometrical proof of the persistence of normally hyperbolic submanifolds

Résumé

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.

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hal-00623713 , version 1 (15-09-2011)

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