Arcs, balls and spheres that cannot be attractors in ${\mathbb{R}}^3$

Abstract

For any compact set $K\subseteq {\mathbb{R}}^3$ we define a number $r(K)$ that is either a nonnegative integer or $\infty$. Intuitively, $r(K)$ provides some information on how wildly $K$ sits in ${\mathbb{R}}^3$. We show that attractors for discrete or continuous dynamical systems have finite $r$ and then prove that certain arcs, balls and spheres cannot be attractors by showing that their $r$ is infinite.