On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields
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hdl:2117/336008
Tipus de documentReport de recerca
Data publicació2020-10-02
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Abstract
Motivated by Poincaré's orbits going to infinity in the (restricted) three-body problem (see \cite{poincare} and \cite{chenciner}), 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~\cite{CMP} of Etnyre--Ghrist's contact/Beltrami correspondence~\cite{EG}, and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck \cite{uhlenbeck}. 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 $\alpha$- and $\omega$-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture.
CitacióMiranda, E.; Oms, C.; Peralta-Salas, D. On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields. 2020.
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URL repositori externhttps://arxiv.org/pdf/2010.00564.pdf
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