Renormalization of multidimensional Hamiltonian flows

Abstract

We construct a renormalization operator acting on the space of analytic Hamiltonians defined on T *Tᵈ, 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 nondegeneracy condition. This in turn yields a new proof of a KAM-type theorem on the stability of diophantine invariant tori.