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Spectral spread and non-autonomous Hamiltonian diffeomorphisms

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For any symplectic manifold \({(M,\omega )}\), the set of Hamiltonian diffeomorphisms \({{\text {Ham}}^c(M,\omega )}\) forms a group and \({{\text {Ham}}^c(M,\omega )}\) contains an important subset \({{\text {Aut}}(M,\omega )}\) which consists of time one flows of autonomous(time-independent) Hamiltonian vector fields on M. One might expect that \({{\text {Aut}}(M,\omega )}\) is a very small subset of \({{\text {Ham}}^c(M,\omega )}\). In this paper, we estimate the size of the subset \({{\text {Aut}}(M,\omega )}\) in \({C^{\infty }}\)-topology and Hofer’s metric which was introduced by Hofer. Polterovich and Shelukhin proved that the complement \({{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}\) is a dense subset of \({{\text {Ham}}^c(M,\omega )}\) in \({C^{\infty }}\)-topology and Hofer’s metric if \({(M,\omega )}\) is a closed symplectically aspherical manifold where Conley conjecture is established (Polterovich and Schelukhin in Sel Math 22(1):227–296, 2016). In this paper, we generalize above theorem to general closed symplectic manifolds and general conv! ex symplectic manifolds. So, we prove that the set of all non-autonomous Hamiltonian diffeomorphisms \({{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}\) is a dense subset of \({{\text {Ham}}^c(M,\omega )}\) in \({C^{\infty }}\)-topology and Hofer’s metric if \({(M,\omega )}\) is a closed or convex symplectic manifold without relying on the solution of Conley conjecture.

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References

  1. Abouzaid, M., Seidel, P.: An open string analogue of Viterbo functoriality. Geom. Topol. 14, 627–718 (2010)

    Article  MathSciNet  Google Scholar 

  2. Floer, A., Hofer, H., Wysocki, K.: Applications of symplectic homology I. Mathematische Zeitschrift 217(1), 577–606 (1994)

    Article  MathSciNet  Google Scholar 

  3. Freifeld, C.: One-parameter Subgroups do not Fill a Neighborhood of the Identity in an Infinite Dimensional Lie (Pseudo)-Group, Battelle Rencontres. Lectures in Mathematics and Physics, pp. 538–543. Benjamin, New York (1967)

    Google Scholar 

  4. Fukaya, K., Ono, K.: Arnold conjecture and Gromov–Witten invariant. Topology 38(5), 993–1048 (1999)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Ginzburg, V.L., Gürel, B.Z.: On the generic existence of periodic orbits in Hamiltonian dynamics. J. Mod. Dyn. 3, 595–610 (2009)

    Article  MathSciNet  Google Scholar 

  7. Hummel, C.: Gromov’s Compactness Theorem for Pseudo-holomorphic Curves. Progress in Mathematics, vol. 151. Birkhauser, Basel (1997)

    Book  Google Scholar 

  8. Liu, G., Tian, G.: Floer homology and Arnold conjecture. J. Differ. Geom. 49(1), 1–74 (1998)

    Article  MathSciNet  Google Scholar 

  9. Milnor, J.: Remarks on infinite-dimensional Lie groups. relativity, groups and topology, COURSE 10

  10. Polterovich, L., Schelukhin, E.: Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules. Sel. Math. 22(1), 227–296 (2016)

    Article  MathSciNet  Google Scholar 

  11. 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  Google Scholar 

  12. Schwarz, M.: On the action spectrum for closed symplectically aspherical manifolds. Pac. J. Math. 193(2), 419–461 (2000)

    Article  MathSciNet  Google Scholar 

  13. Usher, M.: Hofer’s metrics and boundary depth. Ann. Sci. Éc. Norm. Supér. (4) 46(1), 57–128 (2013)

    Article  MathSciNet  Google Scholar 

  14. Viterbo, C.: Functors and computations in Floer homology with applications Part 1. Geom. Funct. Anal. 9(5), 985–1033 (1999)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks his supervisor, Professor Kaoru Ono, for many useful comments, discussions and encouragement. The author is supported by JSPS Research Fellowship for Young Scientists No. 201601854. The author also thanks referees for fruitful suggestions, especially for pointing out an error in our manuscript.

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Correspondence to Yoshihiro Sugimoto.

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Sugimoto, Y. Spectral spread and non-autonomous Hamiltonian diffeomorphisms. manuscripta math. 160, 483–508 (2019). https://doi.org/10.1007/s00229-018-1078-0

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