Arnold Diffusion in Arbitrary Degrees of Freedom and 3-dimensional Normally Hyperbolic Invariant Cylinders

%#(!, #, *) = %0(#) + +%1(!, #, *), * ∈ ! = R/!. We study Arnold diffusion for this system, namely, existence of orbits {(!, #)(*)}$ such that ∣#(*)− #(0)∣ > .(1) independently of +. We say that %0 has a resonance of order / < & at a point # ∈ ( if there are / linearly independent integer vectors 11, . . . , 1% ∈ Z such that 1& ⋅ ∇%0(#) = 0 for 2 = 1, ⋅ ⋅ ⋅ ,/. We say that a resonance is of co-dimension 3 if it is of order &−3. Due to the theorem on implicit function and convexity of %0 a resonance of codimension 3 (if non empty) locally defines a surface of dimension 3. We would like to study dynamics near a resonance of codimension one, i.e. near a segment in (. For any resonance of codimension one there is an integer linear symplectic transformation which brings integer vectors 11, . . . , 1!−1 ∈ Z, defining the resonance, to the form 1& = (0, ⋅ ⋅ ⋅ , 1&, 0, ⋅ ⋅ ⋅ , 0). Since we are interested in a local property assume that a resonance, denoted Γ, of codimension one is of the following form: