3-manifolds which are orbit spaces of diffeomorphisms

In a very general setting, we show that a 3-manifold obtained as the orbit space of the basin of a topological attractor is either $S^2\times S^1$ or irreducible.

We then study in more detail the topology of a class of 3-manifolds which are also orbit spaces and arise as invariants of gradient-like diffeomorphisms (in dimension 3). Up to a finite number of exceptions, which we explicitly describe, all these manifolds are Haken and, by changing the diffeomorphism by a finite power, all the Seifert components of the Jaco–Shalen–Johannson decomposition of these manifolds are made into product circle bundles.