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We prove a form of Arnold diffusion in the a priori stable case. Let H0(p) +$\epsilon$H1($\theta$, p, t), $\theta$ $\in$ T n , p $\in$ B n , t $\in$ T =R/T be a nearly integrable system of arbitrary degrees of freedom n 2 with astrictly convex H0. We show that for a "generic" $\epsilon$H1, there exists anorbit ($\theta$, p)(t) satisfying p(t) -- p(0) {\textgreater} l(H1){\textgreater} 0, where l(H1) is independent of $\epsilon$. The diffusion orbittravels along a co-dimension one resonance , and the only obstruction to ourconstruction is a finite set of additional resonances. For the proof we use acombination geometric and variational methods, and manage to adapt tools whichhave recently been developed in the a priori unstable case.

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