Inspired by Lorenz' remarkable chaotic flow, we describe in this paper the structure of all C 1 robust transitive sets with singularities for flows on closed 3-manifolds: they are partially hyperbolic with volume-expanding central direction, and are either attractors or repellers. In particular, any C 1 robust attractor with singularities for flows on closed 3-manifolds always has an invariant foliation whose leaves are forward contracted by the flow, and has positive Lyapunov exponent at every orbit, showing that any C 1 robust attractor resembles a geometric Lorenz attractor.

Abstract. We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two different strong senses. First, the flow is expansive: if two points remain close at all times, possibly with time reparametrization, then their orbits coincide. Second, there exists a physical (or Sinai-RuelleBowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a u-Gibbs state and is an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strongunstable direction.This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows.In particular these results can be applied (i) to the flow defined by the Lorenz equations, (ii) to the geometric Lorenz flows, (iii) to the attractors appearing in the unfolding of certain resonant double homoclinic loops, (iv) in the unfolding of certain singular cycles and (v) in some geometrical models which are singular-hyperbolic but of a different topological type from the geometric Lorenz models. In all these cases the results show that these attractors are expansive and have physical measures which are u-Gibbs states.

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Link to this article: http://journals.cambridge.org/abstract_S0143385702001621How to cite this article: C. A MORALES and M. J PACIFICO (2003). A dichotomy for three-dimensional vector elds.Abstract. We prove that a generic C 1 vector field on a closed 3-manifold either has infinitely many sinks or sources or else is singular Axiom A without cycles. Singular Axiom A means that the non-wandering set of the vector field has a decomposition into compact invariant sets, each being either a hyperbolic basic set or a singular hyperbolic attractor (like the Lorenz-like ones) or a singular hyperbolic repeller. An attractor is a transitive set which attracts all nearby future orbits, and a repeller is an attractor for the time-reversed flow. Our result implies that generic C 1 vector fields on closed 3-manifolds do exhibit either attractors or repellers.

Abstract. We construct a class of vector fields on 3-manifolds containing the hyperbolic ones and the geometric Lorenz attractor. Conversely, we shall prove that nonhyperbolic systems in this class resemble the Lorenz attractor: they have Lorenz-like singularities accumulated by periodic orbits and they cannot be approximated by flows with nonhyperbolic critical elements.