Normally Hyperbolic Invariant Laminations and diffusive behaviour for the generalized Arnold example away from resonances

In this paper we study existence of Normally Hyperbolic Invariant Laminations (NHIL) for a nearly integrable system given by the product of the pendulum and the rotator perturbed with a small coupling between the two. This example was introduced by Arnold [1]. Using a separatrix map, introduced in a low dimensional case by Zaslavskii-Filonenko [61] and studied in a multidimensional case by Treschev and Piftankin [51, 52, 55, 56], for an open class of trigonometric perturbations we prove that NHIL do exist. Moreover, using a second order expansion for the separatrix map from [27], we prove that the system restricted to this NHIL is a skew product of nearly integrable cylinder maps. Application of the results from [11] about random iteration of such skew products show that in the proper ε-dependent time scale the push forward of a Bernoulli measure supported on this NHIL weakly converges to an Ito diffusion process on the line as ε tends to zero.