Dynamics of the dominant Hamiltonian, with applications to Arnold diffusion

Abstract

It is well known that instabilities of nearly integrable Hamiltonian systems occur around resonances. Dynamics near resonances of these systems is well approximated by the associated averaged system, called slow system. Each resonance is defined by a basis (a collection of integer vectors). We introduce a class of resonances whose basis can be divided into two well separated groups and call them dominant. We prove that the associated slow system can be well approximated by a subsystem given by one of the groups, both in the sense of the vector field and weak KAM theory. One of crucial ingredients of proving Arnold diffusion is understanding the structure of invariant (Aubry) sets of nearly integrable systems. As an important application we construct a diffusion path for a generic nearly integrable system such that invariant (Aubry) sets along this path have a "simple" structure similar to the structure of Aubry-Mather sets of twist maps. This is a crucial ingredient in proving Arnold diffusion for convex Hamiltonians in any number of degrees