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 one-parameter family of plugs $\mathbb{K}_{\unicode[STIX]{x1D716}}$ for $\unicode[STIX]{x1D716}\in (-a,a)$ and $a<1$, such that: (1) the plug $\mathbb{K}_{0}=\mathbb{K}$ is a generic Kuperberg plug; (2) for $\unicode[STIX]{x1D716}<0$, the flow in the plug $\mathbb{K}_{\unicode[STIX]{x1D716}}$ 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 $\unicode[STIX]{x1D716}>0$, the flow in the plug $\mathbb{K}_{\unicode[STIX]{x1D716}}$ has positive topological entropy, and an abundance of periodic orbits.