Symplectic Homogenization

TL;DR

This paper introduces a method called symplectic homogenization for Hamiltonians on cotangent bundles, proving convergence of homogenized sequences and linking the effective Hamiltonian to Mather's alpha function, with implications for Hamilton-Jacobi equations.

Contribution

It establishes the convergence of homogenized Hamiltonian sequences in the gamma topology and connects the effective Hamiltonian to Mather's alpha function, providing new insights into symplectic invariance.

Findings

Proves convergence of $H_k$ to $ar{H}$ in gamma topology.

Shows $ar{H}$ coincides with Mather's alpha function.

Provides applications to Hamilton-Jacobi equations and quasi-states.

Abstract

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 A previous version of this paper relied on the former "On the capacity of Lagrangians in $T^*T^n$ which has been withdrawn.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.