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

Arcs, balls and spheres that cannot be attractors in $\mathbb {R}^3$

“…327]. It explains why all our examples of nonattracting sets are wild, although one should not be misled to think that no wild set can be an attractor: for instance, the dyadic solenoid is wild but it is an attractor, and it is even possible to construct wild arcs that are attractors [27,Example 38,p. 6177].…”

“…Also, the characterization will generally depend on M and whether one is interested in discrete or continuous dynamical systems. The latter are well understood [11,27], so we shall restrict ourselves to discrete dynamical systems.…”

Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems

“…Let M be a 3-manifold. We say that a compactum K ⊂ M is tame if there exists a triangulation (S, h) of M and a subcomplex T ⊂ S such that (T , h| K ) is a triangulation of K. A useful criterion for tameness is [22,Lemma 5] which establishes that if M is a 3-manifold with boundary…”

“…In particular, Ȟ * (K; Z) is finitely generated in every dimension. In spite of this K is not a topological spine of any compact 3-manifold with boundary N ⊂ R 3 since if it were [22,Theorem 4] would ensure the existence of a flow in R 3 having K as an attractor. However it follows from [23, Example 47] that K cannot be an attractor of a flow in R 3 .…”

On the set of wild points of attracting surfaces in R3

“…The answer depends on the dimension n of the ambient space, on whether K is given in the abstract or already as a subset of R n , on whether one considers global or local attractors, continuous or discrete dynamics, etc. There are numerous papers in the literature that deal with this question in some variant or another, among which we may cite [5], [7], [9], [11], [12], [13], [14], [15], [22] or [23]. Here we shall concentrate on the realizability problem for closed surfaces S ⊆ R 3 as attractors for homeomorphisms.…”