Weak K.A.M. solutions and minimizing orbits of twist maps

TL;DR

This paper establishes a Lipschitz-continuous dependence of weak K.A.M. solutions on cohomology classes for exact symplectic twist maps, linking various theories and providing detailed descriptions of minimizing orbits and pseudographs.

Contribution

It introduces a Lipschitz-continuous selection of weak K.A.M. solutions and connects weak K.A.M., Aubry-Mather, and pseudo-foliation theories for twist maps.

Findings

Aubry-Mather sets are contained in vertically ordered pseudographs.

Each vertical of the annulus has at most two points with minimizing negative orbits for a given rotation number.

All pseudographs are filled with minimizing semi-orbits, with a detailed description of a smaller union containing all minimizing orbits.

Abstract

For exact symplectic twist maps of the annulus, we etablish a choice of weak K.A.M. solutions $u_c=u(\cdot, c)$ that depend in a Lipschitz-continuous way on the cohomology class $c$. This allows us to make a bridge between weak K.A.M. theory of Fathi, Aubry-Mather theory for semi-orbits as developped by Bangert and existence of backward invariant pseudo-foliations as seen by Katnelson \& Ornstein. We deduce a very precise description of the pseudographs of the weak K.A.M. solutions and many interesting results as --the Aubry-Mather sets are contained in pseudographs that are vertically ordered by their rotation numbers; --on every image of a vertical of the annulus, there is at most two points whose negative orbit is minimizing with a given rotation number; --all the corresponding pseudographs are filled by minimizing semi-orbits and we provide a description of a smaller selection of full pseudographs whose union contains all the minimizing orbits; --there exists an exact symplectic twist map that has a minimizing negative semi-orbit that is not contained in the pseudograph of a weak K.A.M. solution.