Approximation of ω-limit sets of diffeomorphisms by periodic orbits

Let M be a compact manifold, D the set of its C-1-diffeomorphisms (possibly symplectic or volume preserving). We prove that there exists a dense G(delta) G of D such that if f is an element of G, every omega-limit set of f is the limit (for the Hausdorff topology) of a sequence of periodic orbits. This has certain interesting consequences concerning the structure of the omega-limit sets. Moreover, we define a new notion of attractors and describe them precisely in different cases. (C) 2003 tditions scientifiques et medicales Elsevier SAS.