A counterexample to the singular Weinstein conjecture

Abstract

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 [20]. Our construction shows that there are b- contact manifolds with no singular periodic orbits and no regular periodic orbits 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.