A geometrical proof of the persistence of normally hyperbolic submanifolds

TL;DR

This paper offers a straightforward, geometrical proof demonstrating that normally hyperbolic invariant submanifolds persist under small perturbations, extending classical results with new insights on topologically normally hyperbolic submanifolds.

Contribution

It introduces a simple, computation-free geometric proof of the persistence of normally hyperbolic submanifolds, including new results on topologically normally hyperbolic cases without requiring uniqueness.

Findings

Persistence of Lipschitz invariant submanifolds shown via Schauder fixed point theorem

Smoothness and uniqueness derived through geometric arguments

New results on persistence and regularity of topologically 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 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.