A strong form of Arnold diffusion for three and a half degrees of freedom

We present key elements of a proof of a strong form of Arnold diffusion for systems of three and a half degrees of freedom. More exactly, let T3 be a 3-dimensional torus and B3 be the unit ball around the origin in R3. Fix ρ > 0. Our main result says that for a “generic” time-periodic perturbation of an integrable system of three degrees of freedom H0(p) + εH1(θ, p, t), θ ∈ T3, p ∈ B3, t ∈ T = R/Z, with a strictly convex H0, there exists a ρ-dense orbit (θε, pε, t)(t) in T3×B3×T, namely, a ρ-neighborhood of the orbit contains T3 ×B3 × T. Our proof is a combination of geometric and variational methods and closely related to our previous result [19] on Arnold diffusion from two and a half degrees of freedom We build a “connected” net of 3-dimensional “minimal” normally hyperbolic invariant cylinders. To construct diffusing orbits along this net we employ a version of Mather variational method [20, 21] equipped with weak KAM theory [13] and a notion of cequivalence proposed by Bernard in [5]. 0.1 Statement of the result Let (θ, p) = (θ1, θ2, θ3, p1, p2, p3) ∈ T3 × B3 be the phase space of an integrable Hamiltonian system H0(p) with T3 being a 3-dimensional torus R3/Z3 and B3 being the unit ball around 0 in R3. Assume that H0 is strictly convex, i.e. the Hessian ∂2 pipjH0 is strictly positive definite. Consider a smooth time periodic perturbation Hε(θ, p, t) = H0(p) + εH1(θ, p, t), t ∈ T = R/T. (1) We study a strong form of Arnold diffusion for this system. Namely, for any pair of open sets U,U ′ ∈ B3 we investigate existence of orbits {(θε, pε)(t)}t going from U to U ′, i.e. pε(0) ∈ U and pε(t) ∈ U ′ for some t = tε > 0. ∗University of Maryland at College Park (vadim.kaloshin gmail.com) †University of Toronto (kzhang math.utoronto.edu)