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

TL;DR

The paper investigates the structure of wild points on attracting surfaces in three-dimensional space, proving they are line-like under certain conditions and constructing wild spheres that cannot be attractors.

Contribution

It establishes a topological restriction on wild points of attracting surfaces and constructs examples of wild spheres that are not attractors.

Findings

Wild points are contained in a line if totally disconnected.

Existence of uncountably many wild spheres that are not attractors.

Topological constraints on attracting surfaces in $\

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.