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.
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/
[AI16] - “A closing lemma for Hamiltonian diffeomorphisms of closed surfaces”, Geom. Funct. Anal. 26 (2016) no. 5, p. 1245-1254 | DOI | MR | Zbl
[Bec20] - “Asymptotic action and asymptotic winding number for area-preserving diffeomorphisms of the disk”, 2020 | arXiv
[Bra15] - “Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves”, Ann. of Math. (2) 181 (2015) no. 3, p. 1033-1086 | DOI | MR | Zbl
[Cal70] - “On the group of automorphisms of a symplectic manifold”, in Problems in analysis (Sympos. in honor of Salomon Bochner, Princeton, NJ, 1969), Princeton Univ. Press, Princeton, NJ, 1970, p. 1-26 | Zbl
[CGHS20] - “Proof of the simplicity conjecture”, 2020 | arXiv
[CGPZ21] - “Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms”, 2021 | arXiv
[Cha84] - “Une idée du type « géodésiques brisées »pour les systèmes hamiltoniens”, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984) no. 13, p. 293-296 | Zbl
[EH21] - “PFH spectral invariants and closing lemmas”, 2021 | arXiv
[Fat80] - Transformations et homéomorphismes préservant la mesure; Systèmes dynamiques minimaux, Ph. D. Thesis, Université d’Orsay, 1980
[Fra88] - “Generalizations of the Poincaré-Birkhoff theorem”, Ann. of Math. (2) 128 (1988) no. 1, p. 139-151, Erratum: Ibid., 164 (2006), no. 3, p. 1097–1098 | DOI | Zbl
[GG97] - “Enlacements asymptotiques”, Topology 36 (1997) no. 6, p. 1355-1379 | DOI | MR | Zbl
[Hut16] - “Mean action and the Calabi invariant”, J. Modern Dyn. 10 (2016), p. 511-539 | DOI | MR | Zbl
[Jol21] - About barcodes and Calabi invariant for Hamiltonian homeomorphisms of surfaces, Ph. D. Thesis, Sorbonne Université, 2021
[LC99] - Décomposition des difféomorphismes du tore en applications déviant la verticale, Mém. Soc. Math. France (N.S.), vol. 79, Société Mathématique de France, Paris, 1999 | DOI | Numdam
[LC16] - “A finite dimensional approach to Bramham’s approximation theorem”, Ann. Inst. Fourier (Grenoble) 66 (2016) no. 5, p. 2169-2202 | DOI | Numdam | MR | Zbl
[Pir21] - “Hutchings’s inequality for the Calabi invariant revisited with an application to pseudo-rotations”, 2021 | arXiv
[Sch57] - “Asymptotic cycles”, Ann. of Math. (2) 66 (1957), p. 270-284 | DOI | MR
[She15] - “‘Enlacements asymptotiques’ revisited”, Ann. Math. Qué. 39 (2015) no. 2, p. 205-208 | DOI | MR | Zbl
Cité par Sources :