On the sharpness of the $C^1$-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective

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 \(C^1\) pert