A geometrical proof of the persistence of normally hyperbolic submanifolds

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 subm...

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Bibliographic Details
Published in:Dynamical systems (London, England) England), 2013-12, Vol.28 (4), p.567-581
Main Authors: Berger, Pierre, Bounemoura, Abed
Format: Article
Language:English
Subjects:
Dynamical Systems
invariant manifolds
Invariants
Mathematics
normal hyperbolicity
Proving
Regularity
Smoothness
topological normal hyperbolicity
Transforms
Uniqueness
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Summary: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.
ISSN:1468-9367
1468-9375
DOI:10.1080/14689367.2013.835386