Arcs, balls and spheres that cannot be attractors in R^3

Abstract

For any compact set K ⊆ R3 we define a number r(K) that is either a nonnegative integer or ∞. Intuitively, r(K) provides some information on how wildly K sits in R3. 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.