# Symplectic Homogenization

Claude Viterbo (DMA-Ecole Normale Sup\'erieure)

Arxiv ID: 0801.0206•Last updated: 4/13/2022

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. A previous version of this
paper relied on the former "On the capacity of Lagrangians in $T^*T^n$ which
has been withdrawn. The present version of Symplectic Homogenization does not
rely on it anymore.

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