On the singular Weinstein conjecture and the existence of escape orbits for b-Beltrami fields
Sur la conjecture de Weinstein singulière et l'existence d'orbites d'échappées pour des champs b-Beltrami
Résumé
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.
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