The closing lemma for generalized recurrence in the plane

Abstract

We prove a version of the Closing Lemma for C r {C^r} vector fields on the plane, r ⩾ 1 r \geqslant 1 , and for a kind of recurrence obtained using the concept of prolongational limit sets. We call it generalized recurrence. Given a nonperiodic point p p in the generalized recurrent set we perturb the vector field in order to get a new vector field arbitrarily close to it, with a closed orbit through p p .