We prove that homoclinic classes for a residual set of C 1 vector fields X on closed n-manifolds are maximal transitive, and depend continuously on periodic orbit data. In addition, X does not exhibit cycles formed by homoclinic classes. We also prove that a homoclinic class of X is isolated if and only if it is Ω-isolated, and it is the intersection of its stable set with its unstable set. All these properties are well known for structural stable Axiom A vector fields.

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We prove a perturbation theorem for the k-jets, k ≥ 2, of the Poincaré map of a closed orbit of the Hamiltonian flow of a Tonelli Hamiltonian H : T * M → R, on a closed manifold M . As a consequence we obtain Mañé generic properties of Hamiltonian and Lagrangian flows.

A homoclinic class of a vector field is the closure of the transverse homoclinic orbits associated to a hyperbolic periodic orbit. An attractor is a transitive set to which every positive nearby orbit converges; likewise, every negative nearby orbit converges to a repeller. It is shown in this paper that a generic C 1 vector field on a closed n-manifold has either infinitely many homoclinic classes, or a finite collection of attractors (or, respectively, repellers) with basins that form an open-dense set. This result gives an approach to use in proving a conjecture by Palis. A proof is also given of the existence of a locally residual subset of C 1 vector fields on a 5-manifold having finitely many attractors and repellers, but infinitely many homoclinic classes.

We study the omega-limit sets ω X (x) in an isolating block U of a singular-hyperbolic attractor for three-dimensional vector fields X. We prove that for every vector field Y close to X the set {x ∈ U : ω Y (x) contains a singularity} is residual in U . This is used to prove the persistence of singularhyperbolic attractors with only one singularity as chain-transitive Lyapunov stable sets. These results generalize well known properties of the geometric Lorenz attractor [GW] and the example in [MPu].