Irrational ellipticity when a convex energy surface has finitely many orbits
Let $n \ge 3$ and let $\Sigma \subset \mathbb{R}^{2n}$ be a smooth compact strictly convex hypersurface with positive definite second fundamental form. Its closed characteristics are the closed integral curves of the one-dimensional kernel of the restriction to $\Sigma$ of the symplectic form $\omega = \sum_j dx_j \wedge dy_j$. A closed characteristic is irrationally elliptic if the derivative of its transverse Poincaré return map is symplectically conjugate to a direct sum $$R(\theta_1) \oplus \cdots \oplus R(\theta_{n-1}), \qquad R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}, \qquad \theta_j/(2\pi) \notin \mathbb{Q};$$ in particular this map is diagonalizable and none of its eigenvalues is a root of unity. On an ellipsoid $\sum_j (x_j^2 + y_j^2)/r_j = 2$ whose ratios $r_i/r_j$, $i \ne j$, are all irrational, there are exactly $n$ closed characteristics and all of them are irrationally elliptic. In $\mathbb{R}^4$, Wang, Hu and Long proved that if there are exactly two closed characteristics, both are irrationally elliptic (2007). In every dimension, Hu and Ou proved that if there are finitely many, at least two are elliptic (2017); Li, Liu and Wang proved that if there are exactly $n$, at least two are irrationally elliptic (2025); and Liu and Ou proved that if there are finitely many and $\Sigma$ is non-degenerate, at least two are irrationally elliptic, and that in $\mathbb{R}^6$, for non-degenerate $\Sigma$ with finitely many, at least three are elliptic and at least two of these irrationally elliptic (2026).
Problem. Suppose $\Sigma$ has only finitely many geometrically distinct simple closed characteristics. Must each of them be irrationally elliptic?
Source: Wang, Hu and Long (2007), Conjecture 1.6, for C³ strictly convex hypersurfaces and every n ≥ 2. Stated by AIM — Open Applied Problems for smooth hypersurfaces with positive definite second fundamental form and n ≥ 3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- SG
- Keywords
- AIM Open Applied Problems, closed characteristics, convex hypersurfaces, linear stability
References
@article{wang2007,
author = {Wang, Wei and Hu, Xijun and Long, Yiming},
title = {Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces},
journal = {Duke Math. J.},
volume = {139},
number = {3},
year = {2007},
pages = {411--462},
doi = {10.1215/S0012-7094-07-13931-0},
eprint = {math/0701608},
archivePrefix = {arXiv},
}
@misc{li2025,
author = {Li, Xiaorui and Liu, Hui and Wang, Wei},
title = {Two irrationally elliptic closed orbits of {R}eeb flows on the boundary of star-shaped domain in $\mathbb{R}^{2n}$},
year = {2025},
eprint = {2510.06597},
archivePrefix = {arXiv},
}
@article{hu2017,
author = {Hu, Xijun and Ou, Yuwei},
title = {Stability of closed characteristics on compact convex hypersurfaces in $\mathbb{R}^{2n}$},
journal = {J. Fixed Point Theory Appl.},
volume = {19},
number = {1},
year = {2017},
pages = {585--600},
doi = {10.1007/s11784-016-0366-0},
}
@misc{liu2026,
author = {Liu, Lu and Ou, Yuwei},
title = {Three elliptic closed characteristics on the non-degenerate compact convex hypersurfaces in $\mathbb{R}^6$},
year = {2026},
eprint = {2603.12656},
archivePrefix = {arXiv},
}
@misc{aim-finite-convex-orbits-irrational-ellipticity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Irrational ellipticity when a convex energy surface has finitely many orbits}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/133-finite-convex-orbits-irrational-ellipticity.md},
note = {Entry 133 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-00518, "Irrational ellipticity when a convex energy surface has finitely many orbits", clopen.solutions.