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On the existence of infinitely many invariant Reeb orbits

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Abstract

In this article we extend results of Grove and Tanaka (Bull Am Math Soc 82:497–498, 1976, Acta Math 140:33–48, 1978) and Tanaka (J Differ Geom 17:171–184, 1982) on the existence of isometry-invariant geodesics to the setting of Reeb flows and strict contactomorphisms. Specifically, we prove that if M is a closed connected manifold with the property that the Betti numbers of the free loop space \( \Lambda (M)\) are asymptotically unbounded then for every fibrewise star-shaped hypersurface \(\Sigma \subset T^*M\) and every strict contactomorphism \( \varphi :\Sigma \rightarrow \Sigma \) which is contact-isotopic to the identity, there are infinitely many invariant Reeb orbits.

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Notes

  1. The fact that one takes \(-\theta \) in the definition of \(c_0\) is due to our sign conventions.

  2. The fact that one takes \(-\theta \) in the definition of \(c_0\) is due to our sign conventions.

References

  1. Albers, P., Frauenfelder, U.: Leaf-wise intersections and Rabinowitz Floer homology. J. Topol. Anal. 2(1), 77–98 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Albers, P., Frauenfelder, U.: Spectral Invariants in Rabinowitz Floer homology and Global Hamiltonian perturbations. J. Mod. Dyn. 4, 329–357 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Albers, P., Frauenfelder, U.: Infinitely many leaf-wise intersection points on cotangent bundles. In: Global Differential Geometry, Proceedings in Mathematics, vol. 17, Springer, pp. 437–461 (2012)

  4. Albers, P., Frauenfelder, U.: Rabinowitz Floer homology: A Survey. In: Global Differential Geometry, vol. 17, Proceedings in Mathematics, no. 3, Springer, pp. 437–461 (2012)

  5. Albers, P., Merry, W.J.: Translated points and Rabinowitz Floer homology. J. Fixed Point Theory Appl. 13(1), 201–214 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Abbondandolo, A., Merry, W.J.: Floer homology on the time-energy extended phase space. arXiv:1411.4669 (2014)

  7. Abbondandolo, A., Schwarz, M.: On the Floer homology of cotangent bundles. Commun. Pure Appl. Math. 59, 254–316 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Abbondandolo, A., Schwarz, M.: Estimates and computations in Rabinowitz-Floer homology. J. Topol. Anal. 1(4), 307–405 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bae, Y., Frauenfelder, U.: Continuation homomorphism in Rabinowitz Floer homology for symplectic deformations. Math. Proc. Camb. Philos. Soc. 151, 471–502 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Biran, P., Polterovich, L., Salamon, D.: Propagation in Hamiltonian dynamics and relative symplectic homology. Duke Math. J. 119, 65–118 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cieliebak, K., Frauenfelder, U.: A Floer homology for exact contact embeddings. Pac. J. Math. 239(2), 216–251 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cieliebak, K., Floer, A., Hofer, H., Wysocki, K.: Applications of symplectic homology II: stability of the action spectrum. Math. Z. 223, 27–45 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cieliebak, K., Frauenfelder, U., Oancea, A.: Rabinowitz Floer homology and symplectic homology. Ann. Inst. Fourier 43(6), 957–1015 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Cieliebak, K., Frauenfelder, U., Paternain, G.P.: Symplectic topology of Mañé’s critical values. Geom. Topol. 14, 1765–1870 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ginzburg, V., Gürel, B.: Local Floer homology and the action gap. J. Symplectic Geom. 8(3), 323–327 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ginzburg, V.: On closed trajectories of a charge in a magnetic field. An application of symplectic geometry, Contact and symplectic geometry (Cambridge, 1994) (C. B. Thomas, ed.), Publications of the Newton Institute, vol. 8, , pp. 131–148. Cambridge University Press (1996)

  17. Ginzburg, V.: The Conley conjecture. Ann. Math. 172(2), 1127–1180 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gromoll, D., Meyer, W.: Periodic geodesics on compact riemannian manifolds. J. Differ. Geom. 3, 493–510 (1969)

    MathSciNet  MATH  Google Scholar 

  19. Grove, K.: Condition \(({C})\) for the energy integral on certain path spaces and applications to the theory of geodesics. J. Differ. Geom. 8, 207–223 (1973)

    MathSciNet  MATH  Google Scholar 

  20. Grove, K.: Isometry-invariant geodesics. Topology 13, 281–292 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gromov, M.: Homotopical effects of dilatations. J. Differ. Geom. 13, 303–310 (1978)

    MathSciNet  MATH  Google Scholar 

  22. Grove, K., Tanaka, M.: On the number of invariant closed geodesics. Bull. Am. Math. Soc. 82, 497–498 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  23. Grove, K., Tanaka, M.: On the number of invariant closed geodesics. Acta Math. 140, 33–48 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hryniewicz, U., Macarini, L.: Local contact homology and applications. arXiv:1202.3122 (2012)

  25. Lyusternik, L.A., Fet, A.I.: Variational problems on closed manifolds. Doklady Akad. Nauk SSSR (N.S.) 81, 17–18 (1951)

    MathSciNet  Google Scholar 

  26. Lu, G.: Splitting lemmas for the Finsler energy functional on the space of \({H}^1\)-curves. arXiv:1411.3209 (2014)

  27. Maderna, E.: Invariance of global solutions of the Hamilton-Jacobi equation. Bull. Soc. Math. Fr. 130(4), 493–506 (2002)

    MathSciNet  MATH  Google Scholar 

  28. Mazzucchelli, M.: Isometry-invariant geodesics and the fundamental group, Preprint (2014)

  29. Mazzucchelli, M.: On the multiplicity of isometry-invariant geodesics on product manifolds. Alg. Geom. Topol. 14, 135–156 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. McLean, M.: Computatability and the growth rate of symplectic homology. arXiv:1109.4466 (2011)

  31. McLean, M.: Local Floer homology and infinitely many simple Reeb orbits. Alg. Geom. Topol. 12(4), 1901–1923 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Merry, W.J.: On the Rabinowitz Floer homology of twisted cotangent bundles. Calc. Var. Partial Differ. Equ. 42(3–4), 355–404 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Moser, J.: A fixed point theorem in symplectic geometry. Acta Math. 141(1–2), 17–34 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  34. Paternain, G.P., Paternain, M.: Critical values of autonomous Lagrangians. Comment. Math. Helv. 72, 481–499 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sandon, S.: On iterated translated points for contactomorphisms of \(\mathbb{R}^{2n+1}\), Internat. J. Math. 23, no. 2 (2012)

  36. Salamon, D., Weber, J.: Floer homology and the heat flow. GAFA 16, 1050–1138 (2006)

    MathSciNet  MATH  Google Scholar 

  37. Salamon, D., Zehnder, E.: Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Commun. Pure Appl. Math. 45, 1303–1360 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. Tanaka, M.: On the existence of infinitely many isometry-invariant geodesics. J. Differ. Geom. 17, 171–184 (1982)

    MathSciNet  MATH  Google Scholar 

  39. Viterbo, C.: Functors and computations in Floer homology with applications: Part II. Preprint (1996)

  40. Weigel, P.: Orderable contact structures on Liouville-fillable contact manifolds. arXiv:1304.3662 (2013)

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Acknowledgments

We are very grateful to Marco Mazzucchelli for explaining to us why studying strict contactomorphisms is interesting in this setting, and for numerous helpful discussions on generating functions, and to Viktor Ginzburg for his many detailed and useful comments, and in particular for suggesting Theorem 1.7 to us. The first author thanks Alberto Abbondandolo for many discussions about the \(L^{\infty }\)-estimates in Sect. 6.

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  1. Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Will J. Merry & Kathrin Naef

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  1. Will J. Merry
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  2. Kathrin Naef
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Correspondence to Will J. Merry.

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Merry, W.J., Naef, K. On the existence of infinitely many invariant Reeb orbits. Math. Z. 283, 197–242 (2016). https://doi.org/10.1007/s00209-015-1595-4

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