An approximate renormalization for the break-up of invariant tori with three frequencies

Abstract

Abstract Renormalization theory provides a description of the destruction of invariant tori for Hamiltonian systems of 1 1 2 or 2 degrees of freedom, and explains the self-similarity and the universality of the structures observed. A similar theory for higher dimensional Hamiltonian systems has proved elusive. Here we construct an approximate renormalization for a Hamiltonian system with 2 1 2 degrees of freedom analogous to the lower dimensional version of Escande and Doveil. Using this operator we study the critical surface for the “spiral mean” invariant torus. We find that there is no universal fixed point. Instead the renormalization dynamics on the critical surface is a rotation with irrational winding ratio. Implications for the determination of the exact critical surface are discussed.