A finite dimensional proof of a result of Hutchings about irrational pseudo-rotations
Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 837-866.

We prove that the Calabi invariant of a C 1 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.

Nous montrons que l’invariant de Calabi d’une pseudo-rotation irrationnelle de classe C 1 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.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.234
Classification: 37E30, 37E45, 37J11
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

1 Sorbonne Université, Université Paris-Cité, CNRS, IMJ-PRG F-75005, Paris, France & Institut Universitaire de France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@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, Volume 10 (2023), pp. 837-866. doi : 10.5802/jep.234. https://jep.centre-mersenne.org/articles/10.5802/jep.234/

[AI16] M. Asaoka & K. Irie - “A C closing lemma for Hamiltonian diffeomorphisms of closed surfaces”, Geom. Funct. Anal. 26 (2016) no. 5, p. 1245-1254 | DOI | MR | Zbl

[Bec20] D. Bechara - “Asymptotic action and asymptotic winding number for area-preserving diffeomorphisms of the disk”, 2020 | arXiv

[Bra15] B. Bramham - “Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves”, Ann. of Math. (2) 181 (2015) no. 3, p. 1033-1086 | DOI | MR | Zbl

[Cal70] E. Calabi - “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] D. Cristofaro-Gardiner, V. Humilière & S. Seyfaddini - “Proof of the simplicity conjecture”, 2020 | arXiv

[CGPZ21] D. Cristofaro-Gardiner, R. Prasad & B. Zhang - “Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms”, 2021 | arXiv

[Cha84] M. Chaperon - “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] O. Edtmair & M. Hutchings - “PFH spectral invariants and C closing lemmas”, 2021 | arXiv

[Fat80] A. Fathi - Transformations et homéomorphismes préservant la mesure; Systèmes dynamiques minimaux, Ph. D. Thesis, Université d’Orsay, 1980

[Fra88] J. Franks - “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] J.-M. Gambaudo & É. Ghys - “Enlacements asymptotiques”, Topology 36 (1997) no. 6, p. 1355-1379 | DOI | MR | Zbl

[Hut16] M. Hutchings - “Mean action and the Calabi invariant”, J. Modern Dyn. 10 (2016), p. 511-539 | DOI | MR | Zbl

[Jol21] B. Joly - About barcodes and Calabi invariant for Hamiltonian homeomorphisms of surfaces, Ph. D. Thesis, Sorbonne Université, 2021

[LC99] P. Le Calvez - 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] P. Le Calvez - “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] A. Pirnapasov - “Hutchings’s inequality for the Calabi invariant revisited with an application to pseudo-rotations”, 2021 | arXiv

[Sch57] S. Schwartzman - “Asymptotic cycles”, Ann. of Math. (2) 66 (1957), p. 270-284 | DOI | MR

[She15] E. Shelukhin - “‘Enlacements asymptotiques’ revisited”, Ann. Math. Qué. 39 (2015) no. 2, p. 205-208 | DOI | MR | Zbl

Cited by Sources: