On the role of high order resonances in normal forms and in separatrix splitting

We discuss the role of high order resonances in the construction of normal forms for ε-close to integrable Hamiltonian systems. By heuristic considerations based on standard estimates, we show that the remainder of normal forms is dominated by the terms corresponding to the main high order resonances, and we provide a general argument to show that the size of such leading terms is exponentially small in 1/ε. We apply this method to the problem of estimating the splitting of separatrices in resonant perturbed systems.