  In this article, we investigate Reeb dynamics on $b^m$-contact manifolds,
previously introduced in [MiO], which are contact away from a hypersurface $Z$
but satisfy certain transversality conditions on $Z$. The study of these
contact structures is motivated by that of contact manifolds with boundary. The
search of periodic Reeb orbits on those manifolds thereby starts with a
generalization of the well-known Weinstein conjecture. Contrary to the initial
expectations, examples of compact $b^m$-contact manifolds without periodic Reeb
orbits outside $Z$ are provided. Furthermore, we prove that in dimension $3$,
there are always infinitely many periodic orbits on the critical set if it is
compact. We prove that traps for the $b^m$-Reeb flow exist in any dimension.
This investigation goes hand-in-hand with the Weinstein conjecture on
non-compact manifolds having compact ends of convex type. In particular, we
extend Hofer's arguments to open overtwisted contact manifolds that are
$\mathbb R^+$-invariant in the open ends, obtaining as a corollary the
existence of periodic $b^m$-Reeb orbits away from the critical set. The study
of $b^m$-Reeb dynamics is motivated by well-known problems in fluid dynamics
and celestial mechanics, where those geometric structures naturally appear. In
particular, we prove that the dynamics on positive energy level-sets in the
restricted planar circular three-body problem are described by the Reeb vector
field of a $b^3$-contact form that admits an infinite number of periodic orbits
at the critical set.
