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

Arnaud, Marie-Claude; Zavidovique, Maxime

doi:10.1090/tran/9017

Weak K.A.M. solutions and minimizing orbits of twist maps
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by Marie-Claude Arnaud and Maxime Zavidovique;

Trans. Amer. Math. Soc. 376 (2023), 8129-8171

DOI: https://doi.org/10.1090/tran/9017

Published electronically: September 1, 2023

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Abstract:

For exact symplectic twist maps of the annulus, we establish 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 developed 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 are 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.

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