Renormalization of multidimensional Hamiltonian flows

We construct a renormalization operator acting on the space of analytic Hamiltonians defined on T*T-d, d >= 2, based on the multidimensional continued fractions algorithm developed by the authors. We show convergence of orbits of the operator around integrable Hamiltonians satisfying a non-degeneracy condition. This in turn yields a new proof of a KAM-type theorem on the stability of diophantine invariant tori.