Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders

Abstract

We prove a form of Arnold diffusion in the a-priori stable case. Let H0(p)+ϵH1(θ,p,t),θ∈Tn,p∈Bn,t∈T=R/T,be a nearly integrable system of arbitrary degrees of freedom n⩾ 2 with a strictly convex H0. We show that for a “generic” ϵH1, there exists an orbit (θ, p) satisfying ‖p(t)-p(0)‖>l(H1)>0,where l(H1) is independent of ϵ. The diffusion orbit travels along a codimension-1 resonance, and the only obstruction to our construction is a finite set of additional resonances. For the proof we use a combination of geometric and variational methods, and manage to adapt tools which have recently been developed in the a-priori unstable case.