How strange can an attractor for a dynamical system in a 3-manifold look?

The aim of this paper is to characterise those compact subsets KK of 33-manifolds MM that are (stable and not necessarily global) attractors for some flow on MM. We will show that it is the topology of M−KM−K, rather than that of KK, the one that plays a relevant role in this problem.A necessary and sufficient condition for a set KK to be an attractor is that it must be an “almost tame” subset of MM in a sense made precise under the equivalent notions of “weakly tame” and “tamely embedded up to shape”, defined in the paper. These are complemented by a further equivalent condition, “algebraic tameness”, which has the advantage of being checkable by explicit computation.A final section of the paper is devoted to a partial analysis of the same question when one replaces flows by discrete dynamical systems.