A non-perturbative topological persistence theorem for normally hyperbolic invariant manifolds

Floer proved that, under a sufficiently small $C^0$ perturbation, a normally hyperbolic invariant manifold gives rise to a nearby invariant set whose cohomology ring contains a subring isomorphic to that of the original manifold. Using topological conditions, expressed in terms of the so-called covering relations, we generalize this theorem to a non-perturbative context. Our result does not require the existence of an invariant manifold before perturbation, nor does it require the original map to be smooth or invertible. For the proof, we develop an additional structure on the Conley index for maps on vector bundles.