Some constructions of Weinstein manifolds with chaotic Reeb dynamics

Bott‐integrable Reeb flows on 3‐manifolds

“…Together with the result of Paternain mentioned before Proposition 1.7, Theorem 1.9 shows that every contact structure on 𝑆 3 admits a contact form whose Reeb flow has zero topological entropy; this answers a question raised by Côté [14].…”

“…Reeb flows in higher dimensions that are completely integrable with non-degenerate first integrals in the sense just mentioned have zero topological entropy [51,Theorem 2.2]. On the other hand, on all spheres of dimension 2𝑛 + 1 ⩾ 5, there are contact structures all of whose Reeb flows have positive topological entropy; in dimensions ⩾ 7, this was shown by Alves and Meiwes [5], and in dimension 5 by Côté [14]. Thus, at least under this non-degeneracy assumption, an analogue of Theorem 1.9 does not hold for higher-dimensional spheres.…”