Weak K.A.M. solutions and minimizing orbits of twist maps
Résumé
For exact symplectic twist maps of the annulus, we etablish a choice of weak K.A.M. solutions uc=u(⋅,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.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)