Ergodic Theory and Dynamical Systems

A dichotomy for three-dimensional vector fields

a1 Instituto de Matemática, Universidade Federal do Rio de Janeiro C. P. 68.530, CEP 21.945-970, Rio de Janeiro, R. J., Brazil (e-mail:,

Article author query
morales c   [Google Scholar] 
pacifico m   [Google Scholar] 


We prove that a generic C1 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 C1 vector fields on closed 3-manifolds do exhibit either attractors or repellers.

(Received April 23 2001)
(Revised December 18 2002)