2023/10/30 by Josep Fontana-McNally, Fontana-McNally, Josep, Eva Miranda +5
Mathematics · #Advanced Differential Equations and Dynamical Systems #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2310.19918
openalex publication_date 2023/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 α and ω-limits both lie on Z (a generalized singular periodic orbit). This is the content of the generalized Weinstein conjecture.