Abstract:
In this short note, we prove that singular Reeb vector fields associated with generic bb-contact forms on three dimensional manifolds with compact embedded critical surfaces have either (at least) 2N2N or an infinite number of escape orbits, where NN denotes the number of connected components of the critical set. In case where the first Betti number of a connected component of the critical surface is positive, there exist infinitely many escape orbits. A similar result holds in the case of bb-Beltrami vector fields that are not bb-Reeb. The proof is based on a more detailed analysis of the main result in [19].
Josep Fontana-McNally was supported by an INIREC grant of introduction to research financed
under the project “Computational, Dynamical and Geometrical Complexity in Fluid Dynamics”,
Ayudas Fundación BBVA a Proyectos de Investigación Científica 2021. Josep Fontana, Eva
Miranda and Cédric Oms are partially supported by the Spanish State Research Agency grant
PID2019-103849GB-I00 of AEI / 10.13039/501100011033 and by the AGAUR project 2021 SGR
00603. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies
via an ICREA Academia Prize 2021 and by the Alexander Von Humboldt foundation via a
Friedrich Wilhelm Bessel Research Award. Eva Miranda is also supported by the Spanish State
Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and
Units of Excellence in R&D (project CEX2020-001084-M). Eva Miranda and Daniel Peralta-
Salas acknowledge partial support from the grant “Computational, Dynamical and Geometrical
Complexity in Fluid Dynamics”, Ayudas Fundación BBVA a Proyectos de Investigación Científica
2021. Cédric Oms acknowledges financial support from the Margarita Salas postdoctoral contract
financed by the European Union-NextGenerationEU and is partially supported by the ANR grant
“Cosy” (ANR-21-CE40-0002), partially supported by the ANR grant “CoSyDy” (ANR-CE40-
0014). Daniel Peralta-Salas is supported by the grants CEX2019-000904-S, RED2022-134301-T
and PID2019-106715GB GB-C21 funded by MCIN/AEI/10.13039/501100011033.
\Bibitem{FonMirOms23}
\by Josep Fontana-McNally, Eva Miranda, C\'edric Oms, Daniel Peralta-Salas
\paper From $2N$ to Infinitely Many Escape Orbits
\jour Regul. Chaotic Dyn.
\yr 2023
\vol 28
\issue 4-5
\pages 498--511
\mathnet{http://mi.mathnet.ru/rcd1217}
\crossref{https://doi.org/10.1134/S1560354723520039}
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https://www.mathnet.ru/eng/rcd/v28/i4/p498
This publication is cited in the following 3 articles:
Joaquim Brugués, Eva Miranda, Cédric Oms, “The Arnold conjecture for singular symplectic manifolds”, J. Fixed Point Theory Appl., 26:2 (2024)
Josep Fontana-McNally, Eva Miranda, Cédric Oms, Daniel Peralta-Salas, “A counterexample to the singular Weinstein conjecture”, Advances in Mathematics, 458 (2024), 109998
Pau Mir, Eva Miranda, Pablo Nicolás, “Hamiltonian facets of classical gauge theories on E-manifolds”, J. Phys. A: Math. Theor., 56:23 (2023), 235201
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