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arXiv · 1109.3280

A geometrical proof of the persistence of normally hyperbolic submanifolds

Pierre Berger·Abed Bounemoura
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Abstract

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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Pierre Berger, Abed Bounemoura. 2011-09-15. A geometrical proof of the persistence of normally hyperbolic submanifolds. https://arxiv.org/abs/1109.3280

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