Aperiodicity at the 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.