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arXiv · 2209.11544

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

Marie-Claude Arnaud·Maxime Zavidovique
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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.

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Marie-Claude Arnaud, Maxime Zavidovique. 2022-09-23. Weak K.A.M. solutions and minimizing orbits of twist maps. https://arxiv.org/abs/2209.11544

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