Non-contractible periodic orbits in Hamiltonian dynamics on tori

Ryuma Orita
Bulletin of the London Mathematical Society, vol. 49 (2017), no. 4, pp. 571–580
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Overview

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A non-contractible periodic orbit is a closed trajectory that cannot be continuously shrunk to a point in the underlying manifold. This paper studies whether the existence of one such orbit forces infinitely many others on a standard symplectic torus. Earlier results under atoroidality or toroidal monotonicity assumptions do not cover this case.

The result shows that one non-degenerate, non-contractible periodic orbit implies infinitely many non-contractible simple periodic orbits, meaning orbits that are not iterates of shorter ones. The proof uses the topology of the torus to control the action functional in Floer–Novikov theory. This work was carried out during a visit to the University of Augsburg, with fruitful discussions with Urs Frauenfelder.

@article{orita2017noncontractible,
  author        = {Orita, Ryuma},
  title         = {{Non-contractible periodic orbits in Hamiltonian dynamics on tori}},
  journal       = {Bulletin of the London Mathematical Society},
  volume        = {49},
  number        = {4},
  pages         = {571--580},
  year          = {2017},
  doi           = {10.1112/blms.12054},
  eprint        = {1604.06177},
  archivePrefix = {arXiv}
}