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PreprintCC BY 4.0arXiv

On the sharpness of the C1-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective

  • Yujie Huang
  • Junhao Li
  • Lin Wang
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Abstract

The classical normally hyperbolic invariant manifold theorem asserts that a C1 normally hyperbolic invariant manifold persists under sufficiently small C1 perturbations. For a family of standard-like dissipative twist maps with Jacobian determinant λ∈(0,1), we prove that C1 perturbations of Lipschitz seminorm at most (1−√λ)2 preserve a unique C1 invariant graph, including at equality. Under the strict bound, this graph is normally hyperbolic. Conversely, for every prescribed a*>(1−√λ)2 and every rational frequency, we construct a C∞ perturbation of C1-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 C1 but not normally hyperbolic. Thus the closed bound for C1 regularity and the strict bound for normal hyperbolicity are both sharp.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026sharpness,
  title = {{On the sharpness of the $C^1$-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective}},
  author = {Yujie Huang and Junhao Li and Lin Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.04862},
  eprint = {2608.04862},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.04862},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the sharpness of the C^1-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective
AU  - Yujie Huang
AU  - Junhao Li
AU  - Lin Wang
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2608.04862
UR  - https://arxiv.org/abs/2608.04862
AB  - 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∈(0,1), we prove that C^1 perturbations of Lipschitz seminorm at most (1−√λ)^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−√λ)^2 and every rational frequency, we construct a C^∞ 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.
ER  -