Comptes Rendus
no. 5
Ordinary differential equations/Dynamical systems
A higher-dimensional Poincaré–Birkhoff theorem without monotone twist

Presented by: the Editorial Board

Alessandro Fonda1; Antonio J. Ureña2
1 Dipartimento di Matematica e Geoscienze, Università di Trieste, P.le Europa, 1, 34127 Trieste, Italy
2 Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 475-479.

We provide a simple proof for a higher-dimensional version of the Poincaré–Birkhoff theorem, which applies to Poincaré time maps of Hamiltonian systems. These maps are required neither to be close to the identity nor to have a monotone twist.

Nous fournissons une preuve simple d'une version en plusieurs dimensions du théorème de Poincaré–Birkhoff qui s'applique aux applications de Poincaré des systèmes hamiltoniens. Ces applications ne sont tenues, ni d'être proches de l'identité, ni d'avoir une torsion monotone.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.01.023

Alessandro Fonda 1; Antonio J. Ureña 2

1 Dipartimento di Matematica e Geoscienze, Università di Trieste, P.le Europa, 1, 34127 Trieste, Italy
2 Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain 
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Alessandro Fonda; Antonio J. Ureña. A higher-dimensional Poincaré–Birkhoff theorem without monotone twist. Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 475-479. doi : 10.1016/j.crma.2016.01.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.023/

[1] C.C. Conley; E.J. Zehnder The Birkhoff–Lewis fixed point theorem and a conjecture of V.I. Arnold, Invent. Math., Volume 73 (1983), pp. 33-49

[2] I. Ekeland Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990

[3] A. Fonda; A.J. Ureña On the higher dimensional Poincaré–Birkhoff theorem for Hamiltonian flows, 1: the indefinite twist, 2013 www.dmi.units.it/~fonda/2013_Fonda-Urena.pdf (preprint, available online at)

[4] A. Fonda; A.J. Ureña On the higher dimensional Poincaré–Birkhoff theorem for Hamiltonian flows, 2: the avoiding rays condition, 2014 www.dmi.units.it/~fonda/2014_Fonda-Urena.pdf (preprint, available online at)

[5] A. Szulkin A relative category and applications to critical point theory for strongly indefinite functionals, Nonlinear Anal., Volume 15 (1990), pp. 725-739

[6] A. Szulkin Cohomology and Morse theory for strongly indefinite functionals, Math. Z., Volume 209 (1992), pp. 375-418

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