Knots and Solenoids that Cannot Be Attractors of Self-homeomorphisms of ℝ3

Abstract

As a 1st step to understand how complicated attractors for dynamical systems can be, one may consider the following realizability problem: given a continuum $K \subseteq \mathbb{R}^3$, decide when $K$ can be realized as an attractor for a homeomorphism of $\mathbb{R}^3$. In this paper we introduce toroidal sets as those continua $K \subseteq \mathbb{R}^3$ that have a neighbourhood basis comprised of solid tori and, generalizing the classical notion of genus of a knot, give a natural definition of the genus of toroidal sets and study some of its properties. Using these tools we exhibit knots and solenoids for which the answer to the realizability problem stated above is negative.

The geometric index and attractors of homeomorphisms of $\mathbb{R}^3$

“…Some well known attractors, such as solenoids, are toroidal sets. The ultimate motivation for this paper is to analyze the realizability problem for toroidal sets, extending the work initiated in [2].…”

“…Guided by the general ideas just described, in [2] we introduced the genus g(K) of a toroidal set K as a generalization of the classical genus of a knot. Under appropriate circumstances this genus can be computed as g(K) = lim j g(T j ), where {T j } is any neighbourhood basis of solid tori for K and g(T j ) denotes the genus of the knot represented by the torus T j .…”

“…Using these properties we will exhibit examples (in fact, uncountably many different examples) of toroidal sets that cannot be realized as attractors for homeomorphisms. These examples all have genus zero, and so it is not possible to establish their non-attracting nature using our earlier techniques from [2]. We will also characterize those toroidal sets that can be realized as attractors for flows as those that have finite genus and whose self-index is ∼ 1.…”

“…We call these sets toroidal. In [2] we defined the genus of a toroidal set as a generalization of the classical notion of genus from knot theory.Here we introduce the self-geometric index of a toroidal set K, which captures how each torus T i+1 winds inside the previous T i . We use this index in conjunction with the genus to approach the problem of whether a toroidal set can be realized as an attractor for a flow or a homeomorphism of R 3 .…”

“…We call these sets toroidal. In [2] we defined the genus of a toroidal set as a generalization of the classical notion of genus from knot theory.…”