A counter-example to a C2 closing lemma

Abstract

AbstractLet M be a compact manifold that contains a two-dimensional punctured torus. Given p ∈ M and an integer r ≥ 2, there exists X ∈ ∞(M) having non-trivial recurrent trajectories and such that, for some neighbourhood of X|(M−{p}) in r(M−{p}), no Y ∈ has closed orbits.