2022/09/23 by Marie-Claude Arnaud, Arnaud, Marie-Claude, Maxime Zavidovique +1
Mathematics · #Geometric and Algebraic Topology #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2209.11544
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.