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 ${\mathbb K}_ε$ for $ε\in (-a,a)$ and $a<1$, such that: (1) The plug ${\mathbb K}_0 = {\mathbb K}$ is a generic Kuperberg plug; (2) For $ε<0$, the flow in the plug ${\mathbb K}_ε$ 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 $ε> 0$, the flow in the plug ${\mathbb K}_ε$ has positive topological entropy, and an abundance of periodic orbits.

Minor edits and text revisions from version 1. arXiv admin note: text overlap with arXiv:1306.5025