Arcs, balls and spheres that cannot be attractors in ℝ³

Sánchez-Gabites, J.

doi:10.1090/tran/6570

Arcs, balls and spheres that cannot be attractors in $\mathbb {R}^3$
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Trans. Amer. Math. Soc. 368 (2016), 3591-3627 Request permission

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.

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