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ECH
constraints and twist dynamics in the spatial isosceles
three-body problem
Xijun Hu, Lei Liu, Yuwei Ou, Zhiwen Qiao and Pedro A. S. Salomão |
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Geometry & Topology 30 (2026) 3513–3574
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Abstract |
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We study dynamical constraints arising from Embedded Contact Homology (ECH) in the spatial isosceles three-body problem. For energies below the critical level, the dynamics on the energy surface is identified with a Reeb flow on the tight three-sphere. We first obtain quantitative estimates for the Euler orbit, including monotonicity of its transverse rotation number and a strict inequality comparing its action with the contact volume. Combined with the ECH constraints of Reeb flows on the tight three-sphere with two simple periodic orbits, these estimates rule out the two-orbit scenario, thus forcing every compact energy surface below the critical level to have infinitely many periodic orbits. As a second step, this result admits a qualitative dynamical interpretation via the open book decomposition by disk-like global surfaces of section bounded by the Euler orbit. In this setting, the rotation number and the contact volume define a nontrivial twist interval which encodes the relative winding of periodic orbits. Refining Hutchings’ mean action inequality, we prove that every rational number in the interior of the twist interval is realized as the mean relative winding number of two distinct periodic points of the first return map. For energies above the critical level, where the energy surface is noncompact, we prove the existence of infinitely many periodic orbits and infinitely many parabolic trajectories via twist estimates near infinity. |
Keywords
ECH, isosceles three-body problem, open book
decompositions, area-preserving disk maps, Hamiltonian
systems, symplectic dynamics
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Mathematical Subject Classification
Primary: 37J39, 37J46, 37N05, 53D42
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References |
Publication
Received: 1 April 2026
Revised: 17 May 2026
Accepted: 17 June 2026
Published: 12 October 2026
Proposed: Leonid Polterovich
Seconded: Yakov Eliashberg, Jianfeng Lin
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Authors |
| Xijun Hu | |
| School of Mathematics Shandong University Jinan China |
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| Lei Liu | |
| School of Mathematics Shandong University Jinan China |
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| Yuwei Ou | |
| School of Mathematics Shandong University Jinan China |
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| Zhiwen Qiao | |
| School of Mathematics Shandong University Jinan China |
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| Pedro A. S. Salomão | |
| Shenzhen International Center for
Mathematics Southern University of Science and Technology (SUSTech) Shenzhen China |
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| © 2026 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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