On the destruction of invariant lagrangian graphs for conformal symplectic twist maps

In this article we investigate the fragility of invariant Lagrangian graphs for dissipative maps, focusing on their destruction under small perturbations. Inspired by Herman’s work on conservative systems, we prove that all C0-invariant Lagrangian graphs for an integrable dissipative twist maps can be destroyed by perturbations that are arbitrarily small in the C1-ε-topology. This result is sharp, as evidenced by the persistence of C1-invariant graphs under C1-perturbations guaranteed by the normally hyperbolic invariant manifold theorem.