By Berti M., Biasco L., Bolle P.

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Vol. 190, 1998, pp. 491-508. [12] L. Chierchia, G. Gallavotti: Drift and diffusion in phase space, Annales de l’IHP, section Physique Th´eorique, 60, 1994, pp. 1-144; see also Erratum in Vol. 68, 135, 1998. [13] J. Cresson: Conjecture de Chirikov et optimalit´e des exposants de stabilit´e du th´eor`eme de Nekhoroshev, preprint univ. Besan¸con. [14] J. Cresson, C. Guillet: Periodic orbits and Arnold diffusion, preprint univ. Besan¸con. [15] J. P. Marco: Transitions le long des chaˆınes de tores invariants pour les syst`emes hamiltoniens analytiques, Annales I.

Hence the distance from any point of Rl to Λ is not greater than δR . This completes the proof of (a). 2. 2. Let x ∈ Rl . Again Λ = Λ0 + Zx for some x ∈ Λ such that p · x = 1, hence there exists x ∈ x + Λ such that p · x ∈ [0, 1). We have w α x = y + 2 p , Ω = U + 2 p, |p| |p| with y, U ∈ E = [p]⊥ , w = p · x ∈ [0, 1). We shall assume that α > 0 (if α = 0, there is nothing to prove). Let t = w/α, and consider the time interval defined by J = [0, 1/β] if t < 1/β, J = [t − 1/β, t] if t ≥ 1/β. J ⊂ [0, max{1/β, 1/α}], and it is enough to prove that there exists t ∈ J such that d(x , tΩ + Λ0 ) ≤ δR .

K=0 Now S ∩ [−1/2, 1/2] = ∅, hence u + K ≥ −1/2 and −1 − u − K < 0. As a consequence β/α ≥ 1 − u. Since [−R/2|p|, −R/2|p| + 1] ⊆ [−R/2|p|, R/2|p|] intersects Z − q · x, u ≤ −R/2|p| + 1. Therefore β/α ≥ R/2|p|, which is (i). In particular, since |p| ≤ R, α ≤ 2β. 32 Finally there exists a ∈ [−1, 0) ∩ (Z − q · x); q + ap ∈ Λ∗ , and |q + ap|2 = |q|2 + a2 |p|2 ≤ 3R2 /4 + R2 = 7R /4. Hence q + ap ∈ Λ∗√7R/2 . We have |(q + ap) · Ω| = |β + aα| ≤ β, because −1 ≤ a ≤ 0 and α ≤ 2β. This proves (ii). 2 We first prove that the statement is true for l = 1, with a1 = 1/2.