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

TL;DR

This paper introduces a numerical invariant for compact sets in three-dimensional space to analyze their potential as attractors in dynamical systems, proving some shapes cannot be attractors due to infinite invariant values.

Contribution

It defines the invariant r(K) for compact sets in R^3 and demonstrates that certain shapes with infinite r cannot serve as attractors, advancing understanding of attractor geometry.

Findings

Attractors have finite r values.

Certain arcs, balls, and spheres have infinite r.

Shapes with infinite r cannot be attractors.

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.