EXPONENTIALLY SMALL SPLITTING OF SEPARATRICES FOR PERTURBED INTEGRABLE STANDARD-LIKE MAPS

The splitting of separatrices for the standard-like maps F(x, y) = (y, - x + 2 mu y/1 + y(2) + epsilon V'(y)), mu = cosh h, h > 0, epsilon i s an element of R, is measured. For even entire perturbative potential s V(y) = Sigma(n greater than or equal to 2)V(n)y(2n) such that (V) ov er cap(2 pi) not equal 0, where (V) over cap(xi) = Sigma(n greater tha n or equal to 2) V-n xi(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 pi (V) over cap(2 pi )epsilon e(-pi 2/h)[1 + O(h(2))] (epsilon = o(h(6)ln(-1) h), h --> 0()) This formula agrees with the one provided by the Melnikov theory, w hich cannot be applied directly, due to the exponentially small size o f A with respect to h.