[Une preuve en dimension finie d’un résultat de Hutchings sur les pseudo-rotations irrationnelles]
Nous montrons que l’invariant de Calabi d’une pseudo-rotation irrationnelle de classe qui coïncide avec une rotation sur le bord, est égal au nombre de rotation. Ce résultat a été démontré il y a quelques années par Michael Hutchings (sous des hypothèses légèrement plus fortes). Alors que la démonstration originale s’inscrit dans le formalisme de l’« Embedded Contact Homology », la preuve que nous donnons utilise les fonctions génératrices et les propriétés dynamiques du flot de gradient associé.
We prove that the Calabi invariant of a pseudo-rotation of the unit disk, that coincides with a rotation on the unit circle, is equal to its rotation number. This result has been shown some years ago by Michael Hutchings (under very slightly stronger hypothesis). While the original proof used Embedded Contact Homology techniques, the proof of this article uses generating functions and the dynamics of the induced gradient flow.
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Keywords: Irrational pseudo-rotation, Calabi invariant, generating function, rotation number, linking number
Mot clés : Pseudo-rotation irrationnelle, invariant de Calabi, fonction génératrice, nombre de rotation, nombre d’enlacement
Patrice Le Calvez 1
CC-BY 4.0
@article{JEP_2023__10__837_0,
author = {Patrice Le Calvez},
title = {A finite dimensional proof of a result of {Hutchings} about irrational pseudo-rotations},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {837--866},
publisher = {\'Ecole polytechnique},
volume = {10},
year = {2023},
doi = {10.5802/jep.234},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.234/}
}
TY - JOUR AU - Patrice Le Calvez TI - A finite dimensional proof of a result of Hutchings about irrational pseudo-rotations JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 837 EP - 866 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.234/ DO - 10.5802/jep.234 LA - en ID - JEP_2023__10__837_0 ER -
%0 Journal Article %A Patrice Le Calvez %T A finite dimensional proof of a result of Hutchings about irrational pseudo-rotations %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 837-866 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.234/ %R 10.5802/jep.234 %G en %F JEP_2023__10__837_0
Patrice Le Calvez. A finite dimensional proof of a result of Hutchings about irrational pseudo-rotations. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 837-866. doi : 10.5802/jep.234. https://jep.centre-mersenne.org/articles/10.5802/jep.234/
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