Characterizing omega-limit sets which are closed orbits

Let X be a vector field in a compact n-manifold M, n 2. Given Σ M we say that q M satisfies (P)Σ if the closure of the positive orbit of X through q does not intersect Σ, but, however, there is an open interval I with q as a boundary point such that every positive orbit through I intersects Σ. Among those q having saddle-type hyperbolic omega-limit set ω(q) the ones with ω(q) being a closed orbit satisfy (P)Σ for some closed subset Σ. The converse is true for n=2 but not for n 4. Here we prove the converse for n=3. Moreover, we prove for n=3 that if ω(q) is a singular-hyperbolic set [C. Morales, M. Pacifico, E. Pujals, On C1 robust singular transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris Sér. I 26 (1998) 81–86], [C. Morales, M. Pacifico, E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math. (2) 160 (2) (2004) 375–432], then ω(q) is a closed orbit if and only if q satisfies (P)Σ for some Σ closed. This result improves [S. Bautista, Sobre conjuntos hiperbólicos-singulares (On singular-hyperbolic sets), thesis Uiversidade Federal do Rio de Janeiro, 2005 (in Portuguese)] and [C. Morales, M. Pacifico, Mixing attractors for 3-flows, Nonlinearity 14 (2001) 359–378].