High degeneracy of effective Hamiltonian in two dimensions

Abstract

Consider the effective Hamiltonian $\overline{H}(p)$ associated with the mechanical Hamiltonian $H(p,x)=\frac{1}{2}|p{|}^2+V(x)$. We prove that for generic $V$, $\overline{H}$ is piecewise one-dimensional in a dense open set in two dimensions using Aubry–Mather theory.