Mathematics Colloquia and Seminars

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A closing lemma is a statement asserting that one can create a periodic orbit of some kind of diffeomorphism or vector field through a given open set by a small perturbation. Recently C^\infty closing lemmas have been proved for Reeb vector fields on three-manifolds and for area-preserving surface diffeomorphisms. In this talk we will present a quantitative closing lemma for Reeb vector fields on three-manifolds. This puts upper bounds on the size of the perturbation needed to create a periodic orbit with a given upper bound on the period. The proof uses new spectral invariants of contact three-manifolds.