On the set of wild points of attracting surfaces in ${\mathbb{R}}^3$

Abstract

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. Using this result and a modification of the classical construction of a wild sphere due to Antoine 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.