  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
