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From \(2N\) to Infinitely Many Escape Orbits

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Abstract

In this short note, we prove that singular Reeb vector fields associated with generic \(b\)-contact forms on three dimensional manifolds with compact embedded critical surfaces have either (at least) \(2N\) or an infinite number of escape orbits, where \(N\) 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 \(b\)-Beltrami vector fields that are not \(b\)-Reeb. The proof is based on a more detailed analysis of the main result in [19].

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  • 29 October 2023

    The numbering issue has been changed to 4-5.

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ACKNOWLEDGMENTS

The authors are indebted to the valuable comments of the anonymous referees that improved substantially the results, proofs and the presentation of the previous version of this paper.

Funding

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.

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Authors and Affiliations

  1. Laboratory of Geometry and Dynamical Systems, Universitat Politècnica de Catalunya, EPSEB-UPC, Av. Dr. Marañón, 44-50, 08028, Barcelona, Spain

    Josep Fontana-McNally

  2. Laboratory of Geometry and Dynamical Systems, Universitat Politècnica de Catalunya & IMtech, EPSEB-UPC, Av. Dr. Marañón, 44-50, 08028, Barcelona, Spain

    Eva Miranda

  3. Centre de Recerca Matemàtica, CRM , Campus de Bellaterra, Edifici C, 08193, Barcelona, Spain

    Eva Miranda

  4. BCAM Basque Center of Applied Mathematics , Mazarredo Zumarkalea, 14, 48009, Bilbo, Bizkaia, Spain

    Cédric Oms

  5. ICMAT, C. Nicolás Cabrera, 13-15, 28049, Madrid, Spain

    Daniel Peralta-Salas

Authors
  1. Josep Fontana-McNally
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  2. Eva Miranda
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  3. Cédric Oms
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  4. Daniel Peralta-Salas
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Corresponding author

Correspondence to Eva Miranda.

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The authors declare that they have no conflicts of interest.

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MSC2010

53D05, 53D17, 37N05

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Fontana-McNally, J., Miranda, E., Oms, C. et al. From \(2N\) to Infinitely Many Escape Orbits. Regul. Chaot. Dyn. 28, 498–511 (2023). https://doi.org/10.1134/S1560354723520039

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  • DOI: https://doi.org/10.1134/S1560354723520039

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