Knotting Minimal Sets

Abstract

We consider the ways minimal sets of flows in $S^3$ may be embedded. We prove that given any $C^2$ flow on $S^3$ with positive entropy, there is an uncountable collection $\mathcal{M}$ of topologically distinct minimal sets such that for each $M\in \mathcal{M}$ there are infinitely many embedded copies of $M$ in the flow, each copy with a distinct knot type, thus extending work of Franks and Williams for periodic orbits.