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.

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