On the quasi-periodic d-fold degenerate bifurcation

This paper analyses the d-fold degenerate bifurcation of invariant quasi-periodic tori of normal dimension one. It is shown that for integrable families of diffeomorphisms unfolding a d-fold degenerate invariant torus versally, invariant Diophantine tori in the family persist under a small non-integrable perturbation. As a consequence, a subset of positive measure (with respect to an interpolating smooth manifold) of the bifurcation set persists as well. The proof is a consequence of the translated torus theorem of Rüssmann and Herman.