Regions of applicability of Aubry-Mather Theory for non-convex Hamiltonian

Abstract

Herman constructed an autonomous system of two degrees of freedom which says that in non-convex situations, oscillations do happen and Aubry-Mather Theory cannot apply (see the results due to W. F. Chen in 1992). In this paper, it is shown that although the orbits could visit a region far away from the initial point in phase space, they can only exist in some fixed regions in I = (I ₁, I ₂) plane. Moreover, Aubry-Mather Theory can be applied outside the regions.

Keywords

Twist map / Aubry-Mather Theory / Non-convex Hamiltonian

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