On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields

Motivated by Poincare’s orbits going to infinity in the (restricted) three-body problem ´ (see [29] and [7]), 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 a singular counterpart [4] of Etnyre– Ghrist’s contact/Beltrami correspondence [11], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [33]. 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 a- and ¿-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture. E. M. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Eva Miranda and Cedric Oms are supported by the grants reference number MTM2015-69135-P (MINECO/FEDER) ánd reference number 2017SGR932 (AGAUR) and the project PID2019-103849GB-I00 / AEI / 10.13039/501100011033. C. O. has been supported by an FNR-AFR PhD predoctoral grant (project GLADYSS) until October 2nd, 2020 and by a SECTI-Postdoctoral grant financed by Eva Miranda’s ICREA Academia immediately after. D. P.-S. is supported by the grants MTM PID2019-106715GB-C21 (MICINN) and Europa Excelencia EUR2019- 103821 (MCIU). This work is supported in part by the ICMAT–Severo Ochoa grant SEV-2015-0554 and the CSIC grant 20205CEX001.