Symplectic Homogenization

Let H(q,p) be a Hamiltonian on T * n . Under suitable assumptions H, we show that the sequence (H k ) k≥1 defined by H (q,p)=H(kq,p) converges in γ-topology—defined [Vit92]—to an integrable continuous ¯(p). This is extended to case of non-autonomous Hamiltonians, and more general setting which only some variables are homogenized: consider H(kx,y,q,p) prove it has γ-limit ¯(y,q,p), thus yielding “effective Hamiltonian”. The goal this paper convergence above sequences, state first properties homogenization operator, give applications solutions Hamilton-Jacobi equations, construction quasi-states, etc. We also when convex p, function ¯ coincides with Mather’s α [Mat91] associated Legendre dual H. gives new proof—in torus case—of its symplectic invariance discovered P. Bernard [Ber07].