A geometrical proof of the persistence of normally hyperbolic submanifolds

2011/09/15 by Pierre Berger, Berger, Pierre, Abed Bounemoura +1 · 5 citations
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · doi:10.48550/arxiv.1109.3280

openalex publication_date 2011/09/15 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

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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