The Periodic Orbit Conjecture for Steady Euler Flows

• Cardona, Robert ^[1]
  1. [1] Universitat Politècnica de Catalunya

     Universitat Politècnica de Catalunya

     Barcelona, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 2, 2021
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • 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.

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