  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 $L$.
