On the sharpness of the $C^1$-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Huang, Yujie, Li, Junhao, Wang, Lin
Institution: Beijing Institute of Technology
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7197001556