Symplectic Homogenization

Details Symplectic Homogenization Symplectic Homogenization

2007-12-31

arxiv.org/abs/0801.0206v2

Mathematics

Symplectic Geometry

55 pages

Scientific paper

The paper relies crucially on the former "On the capacity of Lagrangians in T^*T^n$ which has been withdrawn. Existence of Symplectic Homogenization is thus for the moment, conjectural. Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $\gamma$ topology defined by the author, to $\bar{H}(p)$. This is extended to the case where only some of the variables are homogenized, that is the sequence $H(kx,y,q,p)$ where the limit is of the type ${\bar H}(y,q,p)$ and thus yields an "effective Hamiltonian". We give here the proof of the convergence, and the first properties of the homogenization operator, and give some immediate consequences for solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that the function $\bar H$ coincides with Mather's $\alpha$ function which gives a new proof of its symplectic invariance proved by P. Bernard.

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