Arnold diffusion for smooth convex systems of two and a half degrees of freedom

In the present note we announce a proof of a strong form of Arnold diffusion for smooth convex Hamiltonian systems. Let T-2 be a 2-dimensional torus and B-2 be the unit ball around the origin in R-2. Fix rho > 0. Our main result says that for a 'generic' time-periodic perturbation of an integrable system of two degrees of freedom H-0(p) + epsilon H-1(theta, p, t), theta is an element of T-2, p is an element of B-2, t is an element of T = R/Z, with a strictly convex H-0, there exists a rho-dense orbit (theta(epsilon), p(epsilon), t)(t) in T-2 x B-2 x T, namely, a rho-neighborhood of the orbit contains T-2 x B-2 x T. Our proof is a combination of geometric and variational methods. The fundamental elements of the construction are the usage of crumpled normally hyperbolic invariant cylinders from [9], flower and simple normally hyperbolic invariant manifolds from [36] as well as their kissing property at a strong double resonance. This allows us to build a 'connected' net of three-dimensional normally hyperbolic invariant manifolds. To construct diffusing orbits along this net we employ a version of the Mather variational method [41] equipped with weak KAM theory [28], proposed by Bernard in [7].