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Spectral spread and non-autonomous Hamiltonian diffeomorphisms

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Abstract

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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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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