Generic properties of 3-dimensional Reeb flows: Birkhoff sections and entropy (Q6582311)

Birkhoff sections provide a classical tool for studying flows on 3-manifolds because when a flow allows such a section, the dynamics can be reduced to the dynamics of the first-return map on the section. Not every flow on a closed 3-manifold admits Birkhoff sections, but many Reeb flows do.\N\NIn this paper the authors use broken book decompositions to examine Reeb flows on closed 3-manifolds. Broken book decompositions are described, for example, in [\textit{V. Colin} et al., Invent. Math. 231, No. 3, 1489--1539 (2023; Zbl 1512.53098]. Their main theorem is as follows: If the Liouville measure of a nondegenerate contact form on a closed 3-manifold can be approximated by periodic Reeb orbits, then the Reeb flow admits a \(\partial\)-strong Birkhoff section. A \(\partial\)-strong Birkhoff decomposition is a type of Birkhoff section subject to a number of additional technical restrictions.\N\NBecause of the equidistribution theorem of \textit{K. Irie} [J. Symplectic Geom. 19 No. 3, 531--566 (2021; Zbl 1479.53086)], this result implies that the set of contact forms whose Reeb flows have a Birkhoff section contains an open and dense set in the \(C^{\infty}\)-topology.\N\NThe authors also show that the set of contact forms whose Reeb flows have positive topological entropy is open and dense in the \(C^{\infty}\)-topology.