Persistence of normally hyperbolic invariant manifolds in the absence of rate conditions

TL;DR

This paper demonstrates that normally hyperbolic invariant manifolds can persist as invariant sets under perturbations without rate conditions, using a nonperturbative topological approach applicable in broad settings.

Contribution

It introduces a method for establishing invariant sets from normally hyperbolic manifolds without requiring hyperbolic rate conditions, expanding applicability in noninvertible and nonorientable contexts.

Findings

Invariant sets can be obtained without rate conditions.

Persistence results hold in nonorientable Banach bundles.

Method applies even when the perturbed set is not a manifold.

Abstract

We consider perturbations of normally hyperbolic invariant manifolds, under which they can lose their hyperbolic properties. We show that if the perturbed map which drives the dynamical system exhibits some topological properties, then the manifold is perturbed to an invariant set. The main feature is that our results do not require the rate conditions to hold after the perturbation. In this case the manifold can be perturbed to an invariant set, which is not a topological manifold. Our method is not perturbative. It can be applied to establish invariant sets within a prescribed neighbourhood also in the absence of a normally hyperbolic invariant manifold prior to perturbation. The work is in the setting of nonorientable Banach vector bundles, without needing to assume invertibility of the map.