Quadratic growth when a Hamiltonian map has infinitely many periodic orbits
Let $(M^{2n}, \omega)$ be a closed connected symplectic manifold and $\phi$ a Hamiltonian diffeomorphism of $M$, the time-one map of a smooth time-periodic Hamiltonian. For $T > 0$ let $P(T) \in \{0, 1, 2, \dots\} \cup \{\infty\}$ be the number of periodic orbits of $\phi$ whose least period is at most $T$, each orbit counted once (iterates of a shorter orbit are not counted again), and set $\log \infty = \infty$.
Albach proved the case of the sphere: an area-preserving homeomorphism of $S^2$ isotopic to the identity has exactly two periodic points or satisfies the inequality below (2025). Ginzburg proved that every Hamiltonian diffeomorphism of a closed symplectically aspherical manifold has infinitely many periodic orbits (2010). Shelukhin proved, for closed monotone manifolds with semisimple quantum homology, that a Hamiltonian diffeomorphism with more contractible fixed points, counted homologically, than the total dimension of the homology has infinitely many periodic points (2022); as Albach, Hryniewicz and Oancea note, his proof gives $\liminf_{T \to \infty} P(T) \log T / T > 0$.
Problem. If $\phi$ has infinitely many periodic orbits, is it always true that $$\liminf_{T \to \infty} \frac{\log P(T)}{\log T} \ge 2 \, ?$$
Source: Albach, Hryniewicz and Oancea (2026), Problem 1 of their contribution to Albach et al., Open problems in billiards and quantitative symplectic geometry, after Albach's conjecture for Reeb flows on closed 3-manifolds (2025).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SG — Symplectic Geometry
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, Hamiltonian diffeomorphisms, periodic orbits
References
@misc{albach2026,
author = {Albach, Bernhard and Barraud, Jean-Fran{\c{c}}ois and Bialy, Misha and Bimmermann, Johanna and Ch{\'a}vez C{\'a}liz, Ana and Damian, Mihai and Deschamps, Lina and Hryniewicz, Umberto and Humili{\`e}re, Vincent and Khesin, Boris and Maier, Levin and Moreno, Agustin and Oancea, Alexandru and Paris-Romaskevich, Olga and Sorrentino, Alfonso and Tabachnikov, Serge},
title = {Open problems in billiards and quantitative symplectic geometry},
year = {2026},
eprint = {2602.12896},
archivePrefix = {arXiv},
}
@misc{albach2025,
author = {Albach, Bernhard},
title = {Quadratic growth of geodesics on the two-sphere},
year = {2025},
eprint = {2508.00147},
archivePrefix = {arXiv},
}
@article{ginzburg2010,
author = {Ginzburg, Viktor L.},
title = {The {C}onley conjecture},
journal = {Ann. of Math. (2)},
volume = {172},
number = {2},
year = {2010},
pages = {1127--1180},
doi = {10.4007/annals.2010.172.1127},
eprint = {math/0610956},
archivePrefix = {arXiv},
}
@article{shelukhin2022,
author = {Shelukhin, Egor},
title = {On the {H}ofer--{Z}ehnder conjecture},
journal = {Ann. of Math. (2)},
volume = {195},
number = {3},
year = {2022},
pages = {775--839},
doi = {10.4007/annals.2022.195.3.1},
eprint = {1905.04769},
archivePrefix = {arXiv},
}
@misc{aim-hamiltonian-periodic-orbit-growth,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Quadratic growth when a Hamiltonian map has infinitely many periodic orbits}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/254-hamiltonian-periodic-orbit-growth.md},
note = {Entry 254 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00528, "Quadratic growth when a Hamiltonian map has infinitely many periodic orbits", clopen.solutions.