Palis's three-dimensional dichotomy conjecture for vector fields

Let $r\geq 1$ and let $M$ be a three-dimensional manifold. A vector field belongs to ${\cal X}^r(M)$, the space of $C^r$ vector fields on $M$, and a homoclinic tangency is a nontransverse intersection of stable and unstable manifolds of a periodic orbit. A singular hyperbolic attractor or singular hyperbolic repeller is a singular-hyperbolic invariant set with the corresponding attracting or repelling property. Palis's conjecture. For any $r\geq 1$ and any three-dimensional manifold $M$, every vector field in ${\cal X}^r(M)$ can be approximated by one which is hyperbolic, or by one which displays a homoclinic tangency, a singular hyperbolic attractor or a singular hyperbolic repeller. This conjecture proposes that hyperbolicity, homoclinic bifurcations and Lorenz-like singular dynamics account for all non-hyperbolic behavior in dimension three. It is open in general, although the $C^1$ case is asserted in the paper's main theorem.