On the existence of infinitely many non-contractible periodic orbits of Hamiltonian diffeomorphisms of closed symplectic manifolds

Ryuma Orita
Journal of Symplectic Geometry, vol. 17 (2019), no. 6, pp. 1893–1927
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Overview

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This paper asks when a single non-contractible periodic orbit forces infinitely many simple periodic orbits on a closed symplectic manifold. It extends the study from standard tori to manifolds whose fundamental group is virtually abelian or an $\mathrm{R}$-group.

For an aspherical symplectic form, the main result applies to a non-degenerate one-periodic orbit with non-torsion homology class, under a toroidal rationality condition and finiteness assumptions on the relevant one-periodic orbits. It guarantees infinitely many non-contractible simple periodic orbits. The paper also establishes versions for monotone and negative monotone manifolds, with further conditions involving augmented action and sums of local indices. Filtered Floer–Novikov homology, together with the assumptions on the fundamental group, makes it possible to control the action values under iteration.

@article{orita2019existence,
  author        = {Orita, Ryuma},
  title         = {{On the existence of infinitely many non-contractible periodic orbits of Hamiltonian diffeomorphisms of closed symplectic manifolds}},
  journal       = {Journal of Symplectic Geometry},
  volume        = {17},
  number        = {6},
  pages         = {1893--1927},
  year          = {2019},
  doi           = {10.4310/JSG.2019.v17.n6.a9},
  eprint        = {1703.01731},
  archivePrefix = {arXiv}
}