On the Closing Lemma problem for the torus

We investigate the open Closing Lemma problem for vector fields on the 2-dimensional torus. The local C^r Closing Lemma is verified for smooth vector fields that are area-preserving at all saddle points. Namely, given such a C^r vector field X, r ≥ 4, with a non-trivially recurrent point p, there exists a vector field Y arbitrarily near to X in the C^r topology and obtained from X by a twist perturbation, such that p is a periodic point of Y. The proof relies on a new result in 1-dimensional dynamics on the nonexistence of semi-wandering intervals of smooth order-preserving circle maps.