On the sharpness of the $C^1$-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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arXiv · 2608.04862

On the sharpness of the $C^1$-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 $C^1$ normally hyperbolic invariant manifold persists under $C^1$ small perturbations. For a family of standard-like dissipative twist maps, we show that the threshold $(1-\sqrt{\lambda})^2$ for the $C^1$-norm of the perturbation is sharp: there exists a $C^\infty$ perturbation $\phi$ with $\|\phi\|_{C^1} = (1-\sqrt{\lambda})^2$ such that the map preserves a unique invariant graph, but this graph possesses non-differentiable points. On the other hand, whenever $\|\phi\|_{C^1} < (1-\sqrt{\lambda})^2$, the $C^1$ normally hyperbolic invariant manifold persists, where $\lambda$ denotes the Jacobian determinant of the map. This provides a critical threshold phenomenon for the persistence of invariant graphs in dissipative twist maps.

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BibTeX RIS

Yujie Huang, Junhao Li, Lin Wang. 2026-08-05. On the sharpness of the $C^1$-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective. https://arxiv.org/abs/2608.04862

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