On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields
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 c...
| Autores: | , , |
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| Tipo de recurso: | informe técnico |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/336008 |
| Acceso en línea: | https://hdl.handle.net/2117/336008 |
| Access Level: | acceso abierto |
| Palabra clave: | Reeb dynamics Morse theory Fluid dynamics Beltrami fields Classificació AMS::70 Mechanics of particles and systems Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | 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. |
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