On the set of wild points of attracting surfaces in R^3

Abstract

Suppose that a closed surface S ⊆ R3 is an attractor, not necessarily global, for a discrete dynamical system. Assuming that its set of wild points W is totally disconnected, we prove that (up to an ambient homeomorphism) it has to be contained in a straight line. As a corollary we show that there exist uncountably many different 2–spheres in R3 none of which can be realized as an attractor for a homeomorphism. 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.