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Aperiodicity at the boundary of chaos


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.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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