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The Periodic Orbit Conjecture for Steady Euler Flows

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Abstract

The periodic orbit conjecture states that, on closed manifolds, the set of lengths of the orbits of a non-vanishing vector field all whose orbits are closed admits an upper bound. This conjecture is known to be false in general due to a counterexample by Sullivan. However, it is satisfied under the geometric condition of being geodesible. In this work, we use the recent characterization of Eulerisable flows (or more generally flows admitting a strongly adapted one-form) to prove that the conjecture remains true for this larger class of vector fields.

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Notes

  1. Especially in three dimensions, it is sometimes required in the definition that the vector field preserves the Riemannian volume. In our discussion, we check that Sullivan–Thurston’s is Beltrami and volume-preserving.

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Acknowledgements

The author is grateful to Daniel Peralta-Salas, who proposed this question during the author’s stay in Madrid for the Workshop on Geometric Methods in Symplectic Topology in December 2019. Thanks to Francisco Torres de Lizaur for useful comments. The author is grateful to the referee for several comments which improved this paper.

Funding

The author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445) via an FPI Grant. The author is partially supported by the grants MTM2015-69135-P/FEDER and PID2019-103849GB-I00/AEI/10.13039/501100011033, and AGAUR Grant 2017SGR932.

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  1. Laboratory of Geometry and Dynamical Systems, Department of Mathematics, Barcelona Graduate School of Mathematics (BGSMath), Universitat Politècnica de Catalunya, Avinguda del Doctor Marañon 44-50, 08028, Barcelona, Spain

    Robert Cardona

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  1. Robert Cardona
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Correspondence to Robert Cardona.

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Cardona, R. The Periodic Orbit Conjecture for Steady Euler Flows. Qual. Theory Dyn. Syst. 20, 52 (2021). https://doi.org/10.1007/s12346-021-00490-w

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