A counterexample to the singular Weinstein conjecture

In this article, we study the dynamical properties of Reeb vector fields on b-contact manifolds. We show that in dimension 3, the number of so-called singular periodic orbits can be prescribed. These constructions illuminate some key properties of escape orbits and singular periodic orbits, which play a central role in formulating singular counterparts to the Weinstein conjecture and the Hamiltonian Seifert conjecture. In fact, we prove that the above-mentioned constructions lead to counterexamples of these conjectures as stated in [23]. Our construction shows that there are b-contact manifolds with no singular periodic orbit and no regular periodic orbit away from Z. We do not know whether there are constructions with no generalized escape orbits whose $\alpha$ and $\omega$-limits both lie on Z (a generalized singular periodic orbit). This is the content of the generalized Weinstein conjecture.