Aperiodicity at the boundary of chaos. (English) Zbl 1409.37042

Hurder, Steven; Rechtman, Ana

Ergodic Theory Dyn. Syst. 38, No. 7, 2683-2728 (2018).

Summary: 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 \(\mathbb{K}_{\epsilon}\) 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.

MSC:

37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37B40	Topological entropy

Keywords:

Kuperberg plug; periodic orbits; topological entropy

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