[Attracteurs de Birkhoff en toutes dimensions]
We extend to higher dimensions the notion of Birkhoff attractor of a dissipative map. We prove that this notion coincides with the classical Birkhoff attractor. We prove that for the dissipative system associated to the discounted Hamilton-Jacobi equation, the graph of the solution is contained in the Birkhoff attractor. The appendix provides instructive counter-examples in the non-Tonelli case. We also study what happens when we perturb a Hamiltonian system to make it dissipative and let the perturbation go to zero. The paper contains two main results on $\gamma $-supports and elements of the $\gamma $-completion of the space of exact Lagrangians. Firstly the $\gamma $-support of a Lagrangian in a cotangent bundle carries the cohomology of the base and secondly given an exact Lagrangian $L$, any Floer theoretic equivalent Lagrangian is the $\gamma $-limit of Hamiltonian images of $L$.
Nous généralisons en dimension supérieure la notion d’attracteur de Birkhoff des applications dissipatives. Nous montrons que celle-ci coïncide avec l’attracteur de Birkhoff classique en dimension $2$. Nous prouvons que pour les systèmes dissipatifs associés à l’équation de Hamilton-Jacobi escomptée pour des hamiltoniens Tonelli, le graphe de la différentielle de la solution est contenu dans l’attracteur de Birkhoff. L’appendice fournit des contre-exemples lorsque le hamiltonien n’est pas Tonelli. Nous étudions aussi la perturbation d’un système hamiltonien de manière à le rendre dissipatif, lorsque la perturbation tend vers $0$. Cet article contient deux résultats sur les supports et les éléments de la complétion de l’espace des lagrangiennes exactes. Tout d’abord, le support d’une lagrangienne compacte d’un fibré cotangent porte la cohomologie de la base, et ensuite, étant donnée une lagrangienne exacte $L$ , toute lagrangienne qui lui est Floer équivalente est la $\gamma $-limite d’images hamiltoniennes de $L$.
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Keywords: Conformally symplectic dynamics, viscosity solutions, Lagrangian submanifolds, spectral distance, $\gamma $-support, nearby Lagrangian conjecture
Mots-clés : Dynamique conformément symplectique, solutions de viscosité, sous-variétés lagrangiennes, distance spectrale, $\gamma $-support, conjecture de la lagrangienne proche
Marie-Claude Arnaud 1 ; Vincent Humilière 2 ; Claude Viterbo 3
CC-BY 4.0
@article{JEP_2026__13__759_0,
author = {Marie-Claude Arnaud and Vincent Humili\`ere and Claude Viterbo},
title = {Higher dimensional {Birkhoff} attractors (with an appendix by {Maxime} {Zavidovique)}},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {759--808},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.336},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.336/}
}
TY - JOUR AU - Marie-Claude Arnaud AU - Vincent Humilière AU - Claude Viterbo TI - Higher dimensional Birkhoff attractors (with an appendix by Maxime Zavidovique) JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 759 EP - 808 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.336/ DO - 10.5802/jep.336 LA - en ID - JEP_2026__13__759_0 ER -
%0 Journal Article %A Marie-Claude Arnaud %A Vincent Humilière %A Claude Viterbo %T Higher dimensional Birkhoff attractors (with an appendix by Maxime Zavidovique) %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 759-808 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.336/ %R 10.5802/jep.336 %G en %F JEP_2026__13__759_0
Marie-Claude Arnaud; Vincent Humilière; Claude Viterbo. Higher dimensional Birkhoff attractors (with an appendix by Maxime Zavidovique). Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 759-808. doi: 10.5802/jep.336
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