On the singular Weinstein conjecture and the existence of escape orbits for b-Beltrami fields
Eva Miranda, Cédric Oms, Daniel Peralta‐Salas
Abstract
Motivated by Poincaré’s orbits going to infinity in the (restricted) three-body problem (see [26] and [6]), we investigate the generic existence of heteroclinic-like orbits in a neighbourhood of the critical set of a b-contact form. This is done by using the singular counterpart [3] of Etnyre– Ghrist’s contact/Beltrami correspondence [9], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [29]. Specifically, we analyze the b-Beltrami vector fields on b-manifolds of dimension 3 and prove that for a generic asymptotically exact b-metric they exhibit escape orbits. We also show that a generic asymptotically symmetric b-Beltrami vector field on an asymptotically flat b-manifold has a generalized singular periodic orbit and at least 4 escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose α- and ω-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture