1. Higher Dimensional Birkhoff attractors (with an appendix by Maxime Zavidovique)
- Author
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Arnaud, Marie-Claude, Humilière, Vincent, and Viterbo, Claude
- Subjects
Mathematics - Symplectic Geometry ,Mathematics - Dynamical Systems ,37J99 (Primary) 37E40, 53D40, 70H20 (Secondary) - 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 $\gamma$-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 $L$. The appendix provides instructive counter-examples., Comment: 42 pages. v2 includes minor corrections and a new appendix by Maxime Zavidovique with instructive counter-examples
- Published
- 2024