Separatrix maps in Hamiltonian systems

This is the authors' abstract. We don't add key points for this paper.

The separatrix map is constructed for some classes of problems in Hamiltonian dynamics. The formulae obtained are used to study two-dimensional symplectic maps close to integrable maps: elliptic periodic trajectories passing through separatrix lobes are constructed, and some estimates for the width of the stochastic layer are given. For Hamiltonian systems with two and a half degrees of freedom it is proved that the Arnol'd diffusion in the a priori unstable case is generic, and in the Mather problem trajectories are constructed for which the mean energy growth is linear in time.