Exponentially small splittings in Hamiltonian systems

Families of area preserving analytical maps, depending on a small parameter epsilon, are considered, with the case epsilon=0 corresponding to an integrable map. The asymptotic formulas for the splittings of separatrices are derived by the method of analytical continuation of the separatrices to the complex domain. The main terms of the asymptotics are exponentially small with respect to the size of the perturbation. As epsilon tends to zero, the intersection angle of the separatrices can oscillate. The exponent and the oscillatory multiplier of the asymptotic formulas are determined by the position of poles of the homoclinic (heteroclinic) orbit of the limiting flow. Pre-exponential coefficients in the asymptotic formulas contain a multiplier obtained by the numerical study of separatrices of "model" maps in the complex domain.