Higher Dimensional Birkhoff attractors

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.