A counter-example to a C2 closing lemma

Abstract

Let M be a compact manifold that contains a two-dimensional punctured torus. Given p [omitted formula] M and an integer r ≥ 2, there exists X [omitted formula]∞(M) having non-trivial recurrent trajectories and such that, for some neighbourhood [omitted formula] of X|(M-{p}) in [omitted formula]r(M-{p}), no Y [omitted formula] has closed orbits. © 1987, Cambridge University Press. All rights reserved.