Work

Higher Dimensional Birkhoff attractors (with an appendix by Maxime Zavidovique)

Year: 2024

Type: preprint

Abstract: 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 a solution is contained in the Birkhoff attractor. 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 important results on $γ$-supports and elements of the $γ$-completion of the space of exact Lagrangians. Firstly the $γ$-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 $γ$-limit of Hamiltonian images of $L$. The appendix provides instructive counter-examples.

Source: HAL (Le Centre pour la Communication Scientifique Directe)

DOI: 10.48550/arxiv.2404.00804

Authors: Arnaud, Marie-Claude, Humilière, Vincent, Viterbo, Claude

Institutions: Agence Nationale de la Recherche, Centre National de la Recherche Scientifique, Université Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, Sorbonne Paris Cité

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Agence Nationale de la Recherche

Award: Conformally symplectic dynamics, beyond symplectic dynamics

OpenAlex ID: W4393724837

Cites:

Cited by: 2

Related to: 10