PreprintCC BY 4.0arXiv
A non-perturbative topological persistence theorem for normally hyperbolic invariant manifolds
Abstract
Floer proved that, under a sufficiently small C0 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.
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BibTeX
@misc{spolnik2026nonperturbative,
title = {{A non-perturbative topological persistence theorem for normally hyperbolic invariant manifolds}},
author = {Wojciech Spólnik and Maciej J. Capiński},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35258},
eprint = {2610.35258},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35258},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A non-perturbative topological persistence theorem for normally hyperbolic invariant manifolds
AU - Wojciech Spólnik
AU - Maciej J. Capiński
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35258
UR - https://arxiv.org/abs/2610.35258
AB - 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.
ER -