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
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.
References
Arnold, V.I., Khesin, B.: Topological Methods in Hydrodynamics. Springer, New York (1999)
Besse, A.L.: Manifolds all of whose geodesics are closed. Volume 93 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer-Verlag, Berlin-New York, 1978. With appendices by D.B.A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J.L. Kazdan
Cardona, R.: Steady Euler flows and Beltrami fields in high dimensions. Ergodic Theory Dyn. Syst. (2020). https://doi.org/10.1017/etds.2020.124
Cieliebak, K., Volkov, E.: A note on the stationary Euler equations of hydrodynamics. Ergod. Theory Dyn. Syst. 37, 454–480 (2017)
Edwards, R., Millett, K., Sullivan, D.: Foliations with all leaves compact. Topology 16, 13–32 (1977)
Epstein, D.B.A.: Periodic flows on 3-manifolds. Ann. Math. 95, 68–82 (1972)
Epstein, D.B.A., Vogt, E.: A counterexample to the periodic orbit conjecture in codimension 3. Ann. Math. 108(3), 539–552 (1978)
Gluck, H.: Dynamical behavior of geodesic fields. Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), Lecture Notes in Mathematics, vol. 819, pp. 190–215. Springer, Berlin (1980)
Morgan, F.: Geometric Measure Theory: A beginner’s Guide, 5th edn. Academic Press, London (2016)
Mounoud, P., Suhr, S.: Pseudo-Riemannian geodesic foliations by circles. Math. Z. 274, 225–238 (2013)
Peralta-Salas, D., Rechtman, A., Torres de Lizaur, F.: A characterization of 3D Euler flows using commuting zero-flux homologies. Ergod. Theory Dyn. Syst. (2020). https://doi.org/10.1017/etds.2020.25
Rechtman, A.: Use and disuse of plugs in foliations. PhD Thesis, ENS Lyon (2009)
Reeb, G.: Sur certaines propiétés topologiques des variétés feuilletées. Actual. scient. ind. 1183 (1952)
Sullivan, D.: A counterexample to the periodic orbit conjecture. Publ. IHES 46, 5–14 (1976)
Sullivan, D.: A foliation of geodesics is characterized by having no “tangent homologies.” J. Pure Appl. Algebra 13(1), 101–104 (1978)
Tao, T.: On the universality of potential well dynamics. Dyn. PDE 14, 219–238 (2017)
Wadsley, A.W.: Geodesic foliations by circles. J. Diff. Geom. 10, 541–549 (1975)
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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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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DOI: https://doi.org/10.1007/s12346-021-00490-w