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

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.