Publication date:
31 July 2017
Source:Advances in Mathematics, Volume 315
Author(s): J.J. Sánchez-Gabites
Suppose that a closed surface $S\subseteq {\mathbb{R}}^3$ 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 ${\mathbb{R}}^3$ 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 $K\subseteq {\mathbb{R}}^3$ and is related to “how wildly” it sits in ${\mathbb{R}}^3$. We establish the topological results that (i) $r(W)\le r(S)$ 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.