{
  "abstract": "The classical normally hyperbolic invariant manifold theorem asserts that a \\(C^1\\) normally hyperbolic invariant manifold persists under sufficiently small \\(C^1\\) perturbations. For a family of standard-like dissipative twist maps with Jacobian determinant \\(λ\\in(0,1)\\), we prove that \\(C^1\\) perturbations of Lipschitz seminorm at most \\((1-\\sqrtλ)^2\\) preserve a unique \\(C^1\\) invariant graph, including at equality. Under the strict bound, this graph is normally hyperbolic. Conversely, for every prescribed \\(a^*>(1-\\sqrtλ)^2\\) and every rational frequency, we construct a \\(C^\\infty\\) perturbation of \\(C^1\\)-norm exactly \\(a^*\\) whose unique invariant graph is Lipschitz but not differentiable everywhere. We also give an analytic perturbation at the critical norm whose invariant graph is \\(C^1\\) but not normally hyperbolic. Thus the closed bound for \\(C^1\\) regularity and the strict bound for normal hyperbolicity are both sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.04862",
  "author_total": 3,
  "authors": [
    {
      "name": "Yujie Huang"
    },
    {
      "name": "Junhao Li"
    },
    {
      "name": "Lin Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2gjg80cbkjcpt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.04862",
  "fulltext_url": null,
  "id": "2gjg80cbkjcpt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.04862",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7197001556",
  "pdf_url": "https://arxiv.org/pdf/2608.04862",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2gjg80cbkjcpt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2608.04862",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.04862"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "On the sharpness of the $C^1$-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective",
  "title_text": "On the sharpness of the C^1-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/2gjg80cbkjcpt",
  "venue": "arXiv",
  "versions": []
}