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It is well known that instabilities of nearly integrable Hamiltonian systemsoccur around resonances. Dynamics near resonances of these systems is wellapproximated by the associated averaged system, called slow system. Eachresonance is defined by a basis (a collection of integer vectors). We introducea class of resonances whose basis can be divided into two well separated groupsand call them dominant. We prove that the associated slow system can be wellapproximated by a subsystem given by one of the groups, both in the sense ofthe vector field and weak KAM theory.  One of crucial ingredients of proving Arnold diffusion is understanding thestructure of invariant (Aubry) sets of nearly integrable systems. As animportant application we construct a diffusion path for a generic nearlyintegrable 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 convexHamiltonians in any number of degrees

Dynamics of the dominant Hamiltonian, with applications to Arnold diffusion