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

Abstract

For any compact set K ⊆ R 3 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 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.

On the set of wild points of attracting surfaces in R3

“…(2) The conditions laid out in Theorem 2 are sufficient, but not necessary, to guarantee that a surface (even a surface with a totally disconnected set of wild points) cannot be an attractor. A suitable example is the horned sphere S of Alexander, which cannot be realized as an attractor (this is proved in [20]) but whose set of wild points is a rectifiable Cantor set because it is contained in a straight line by construction. The Alexander sphere is shown in Figure 3 and described carefully in [6,Chapter 2].…”

“…Proposition 1 is proved in the very brief Section 2, where we also discuss succintly the relation between wildness and tameness. In Section 3 we recall from [20] the definition and properties of certain number r(K) that can be associated to any compact set K ⊆ R 3 and somehow measures its "crookedness" as a subset of Euclidean space. This r(K) is finite in the case of attractors, and it is because of this property that it allows us to uncouple the topological and dynamical arguments underlying the proof of Theorem 2: dynamics will enter the picture only through the condition that r of an attractor is finite whereas the hard work is on the topological side, in Sections 4 and 5.…”

“…This case appears naturally in dynamics; for instance, one may think of the invariant tori that may arise in an action-angle description of a Hamiltonian system. In previous papers we were able to prove that the horned sphere of Alexander cannot be realized as an attractor [20] and, jointly with R. Ortega [18], that the same holds true for the sphere of Antoine. Building on this it was possible to construct surfaces of any genus that cannot be attractors either.…”

“…In [20] we described how to associate, to each compact subset K ⊆ R 3 , a number r(K) that provides a measure of how wildly K sits in R 3 . We used it to show that certain arcs, balls and spheres (for instance, the horned sphere of Alexander in Figure 3.…”

“…Since this number r(K) will also be the fundamental tool in the proof of Theorem 2, we devote this section to review its definition and some of its properties. We refer the reader to [20] for more details and proofs.…”