On the set of wild points of attracting surfaces in R3

Our techniques hinge on a quantity r(K) that can be defined for any compact set K⊆R3 and is related to “how wildly” it sits in R3. We establish the topological results that (i) r(W)≤r(S) and (ii) any totally disconnected set having a finite r must be contained in a straight line (up to an ambient homeomorphism). The main result follows from these and the fact that attractors have a finite r.