Existence of attractors, homoclinic tangencies and singular hyperbolicity for flows
arXiv:1308.1734
Abstract
We prove that every generic three-dimensional flow has either infinitely many sinks, or, infinitely many hyperbolic or singular-hyperbolic attractors whose basins form a full Lebesgue measure set. We also prove in the orientable case that the set of accumulation points of the sinks of a generic three-dimensional flow has no dominated splitting with respect to the linear Poincaré flow. As a corollary we obtain that every three-dimensional flow can be approximated by flows with homoclinic tangencies or by singular-Axiom A flows.