[Orbites périodiques et sections de Birkhoff des structures hamiltoniennes stables]
Les structures hamiltoniennes stables généralisent les formes de contact et définissent un champ de vecteurs préservant une forme de volume, connu sous le nom de champ de vecteurs de Reeb. Nous étudions deux aspects des champs de vecteurs de Reeb définis par des structures hamiltoniennes stables en dimension : d’une part, nous classons tous les exemples avec un nombre fini d’orbites périodiques sous une condition de non-dégénérescence ; d’autre part, nous donnons des conditions suffisantes pour l’existence d’une décomposition en livres brisés et pour l’existence d’une section de Birkhoff.
Stable Hamiltonian structures generalize contact forms and define a volume-preserving vector field known as the Reeb vector field. We study two aspects of Reeb vector fields defined by stable Hamiltonian structures on -manifolds: on the one hand, we classify all the examples with finitely many periodic orbits under a non-degeneracy condition; on the other, we give sufficient conditions for the existence of a supporting broken book decomposition and for the existence of a Birkhoff section.
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Keywords: Stable hamiltonian structure, Birkhoff section, periodic orbit, transverse section
Mots-clés : Structure hamiltonienne stable, section de Birkhoff, orbites périodiques, section transverse
Robert Cardona 1 ; Ana Rechtman 2
CC-BY 4.0
@article{JEP_2025__12__235_0,
author = {Robert Cardona and Ana Rechtman},
title = {Periodic orbits and {Birkhoff} sections of stable {Hamiltonian} structures},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {235--286},
publisher = {\'Ecole polytechnique},
volume = {12},
year = {2025},
doi = {10.5802/jep.289},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.289/}
}
TY - JOUR AU - Robert Cardona AU - Ana Rechtman TI - Periodic orbits and Birkhoff sections of stable Hamiltonian structures JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 235 EP - 286 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.289/ DO - 10.5802/jep.289 LA - en ID - JEP_2025__12__235_0 ER -
%0 Journal Article %A Robert Cardona %A Ana Rechtman %T Periodic orbits and Birkhoff sections of stable Hamiltonian structures %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 235-286 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.289/ %R 10.5802/jep.289 %G en %F JEP_2025__12__235_0
Robert Cardona; Ana Rechtman. Periodic orbits and Birkhoff sections of stable Hamiltonian structures. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 235-286. doi : 10.5802/jep.289. https://jep.centre-mersenne.org/articles/10.5802/jep.289/
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