  In this paper we use broken book decompositions to study Reeb flows on closed
$3$-manifolds. We show that if the Liouville measure of a nondegenerate contact
form can be approximated by periodic orbits, then there is a Birkhoff section
for the associated Reeb flow. In view of Irie's equidistribution theorem, this
is shown to imply that the set of contact forms whose Reeb flows have a
Birkhoff section contains an open and dense set in the $C^\infty$-topology. We
also show that the set of contact forms whose Reeb flows have positive
topological entropy is open and dense in the $C^\infty$-topology.
