A renormalization approach to lower-dimensional tori with Brjuno frequency vectors

We extend the renormalization scheme for vector fields on in order to construct lower-dimensional invariant tori with Brjuno frequency vectors for near-integrable Hamiltonian flows. For every Brjuno frequency vector and every vector satisfying a Diophantine condition with respect to ω, there exists an analytic manifold of infinitely renormalizable Hamiltonian vector fields; each vector field on is shown to have an analytic invariant torus with frequency vector ω.