Knotting Minimal Sets

TL;DR

This work investigates embeddings of one-dimensional minimal sets $M$ in $S^3$, extending knot-theoretic techniques from periodic orbits to general minimal sets, including solenoids and Sturmian systems. It develops flow expansions and handlebody presentations, analyzes the knot group $G(M)=\pi_1(S^3\setminus M)$ with Alexander duality linking $H_1(S^3\setminus M)$ to $\check H^1(M)$, and uses suspensions and templates to generate and classify embeddings. The authors prove that any $C^2$ flow on $S^3$ with positive entropy contains an uncountable collection $\mathcal{M}$ of topologically distinct minimal sets, each occurring with infinitely many embedded copies of distinct knot types, thereby generalizing Franks–Williams for periodic orbits. They also study surface expansions to obtain tractable knot-group computations and demonstrate density of Sturmian minimal sets in suspensions, illustrating the abundance of knotted embeddings in invariant sets. The results provide a unified framework linking entropy, symbolic dynamics, and intricate knotting in invariant 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.

Figures (9)

• Figure 1: Generators and relations. In the middle diagram, a crossing, with our convention, gives the relation $a=cab^{-1}$. The right hand diagram, of the wedge point, gives the relation $ab=dc$.
• Figure 2: Two embeddings of stages for the dyadic solenoid.
• Figure 3: Two embeddings of stages for the Fibonacci minimal set.
• Figure 4: The simplified Thue-Morse embedding.
• Figure 5: Embedding of $R$ in $R$ according to the cases $n_i=1$ (centre) and $n_i=2$ (right).

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Theorem 2.7
• proof
• Proposition 2.8