A geometrical proof of the persistence of normally hyperbolic submanifolds

Abstract

We present a simple, computation-free and geometrical proof of the following classical result: for a diffeomorphism of a manifold, any compact submanifold that 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, we also prove a new result on the persistence and regularity of ‘topologically’ normally hyperbolic submanifolds, but without any uniqueness statement.

Keywords:

• invariant manifolds
• normal hyperbolicity
• topological normal hyperbolicity

Acknowledgements

The authors are grateful to the hospitality of IMPA. This work has been partially supported by the Balzan Research Project of J. Palis at IMPA. We wish to thank the anonymous referee for interesting remarks on this paper.