Aperiodicity at the boundary of chaos

TL;DR

This paper studies how small smooth variations of a specific flow on a Kuperberg plug can lead to drastic changes in dynamical behavior, including the emergence of chaos and many periodic orbits.

Contribution

It demonstrates the existence of a smooth family of flows on Kuperberg plugs that transition from zero entropy to positive entropy, revealing a boundary of chaos.

Findings

For negative perturbations, flows have two periodic orbits and zero entropy.

For positive perturbations, flows exhibit positive entropy and many periodic orbits.

The transition occurs at a specific parameter value, indicating a boundary of chaos.

Abstract

We consider the dynamical properties of $C^{\infty}$-variations of the flow on an aperiodic Kuperberg plug ${\mathbb K}$. Our main result is that there exists a smooth 1-parameter family of plugs ${\mathbb K}_{\epsilon}$ for $\epsilon \in (-a,a)$ and $a<1$, such that: (1) The plug ${\mathbb K}_0 = {\mathbb K}$ is a generic Kuperberg plug; (2) For $\epsilon <0$, the flow in the plug ${\mathbb K}_{\epsilon}$ has two periodic orbits that bound an invariant cylinder, all other orbits of the flow are wandering, and the flow has topological entropy zero; (3) For $\epsilon > 0$, the flow in the plug ${\mathbb K}_{\epsilon}$ has positive topological entropy, and an abundance of periodic orbits.