Exponentially small splitting of separatrices for perturbed integrable standard like maps

The splitting of separatrices for the standard-like maps is measured. For even entire perturbative potentials V (y) = P n2 V n y 2n such that b V (2) 6 = 0, where b V () = P n2 V n 2n?1 =(2n ? 1)! is the Borel transform of V (y), the following asymptotic formula for the area A of the lobes between the perturbed separatrices is established A = 8 b V (2)" e ? 2 =h h 1 + O(h 2) i (" = o(h 6 ln ?1 h); h ! 0 +): This formula agrees with the one provided by the Melnikov theory, which cannot be applied directly, due to the exponentially small size of A with respect to h.