Consider a differentiable mapping having an invariant torus. It is assumed that the torus has both a stable manifold and unstable and central directions. The authors give conditions for the existence and regularity of invariant manifolds under various classes of perturbations (Lipschitz, of class \(C^k\), of class \(C^{\infty}\)).
They also prove a variant of the lambda lemma (stating that if a manifold \(\Gamma\) has a point of transverse intersection with the stable manifold, then the closure of the positive orbit of \(\Gamma\) contains the unstable manifold). This result is an important tool in the study of the so-called Arnold diffusion.