On the transversal dependence of weak K.A.M. solutions for symplectic twist maps

TL;DR

This paper demonstrates the continuous dependence of weak K.A.M. solutions on cohomology classes for symplectic twist maps, characterizes $C^0$ integrability via regularity of solutions, and provides examples of complex foliations.

Contribution

It introduces a continuous selection of weak K.A.M. solutions depending on cohomology, linking Aubry-Mather sets to pseudographs, and characterizes $C^0$ integrability through solution regularity.

Findings

Weak K.A.M. solutions depend continuously on cohomology classes.

Aubry-Mather sets are contained in vertically ordered pseudographs.

Characterization of $C^0$ integrable twist maps via regularity of solutions.

Abstract

For a symplectic twist map, we prove that there is a choice of weak K.A.M. solutions that depend in a continuous way on the cohomology class. We thus obtain a continuous function $u(\theta, c)$ in two variables: the angle $\theta$ and the cohomology class $c$. As a result, we prove that the Aubry-Mather sets are contained in pseudographs that are vertically ordered by their rotation numbers. Then we characterize the $C^0$ integrable twist maps in terms of regularity of $u$ that allows to see $u$ as a generating function. We also obtain some results for the Lipschitz integrable twist maps. With an example, we show that our choice is not the so-called discounted one (see \cite{DFIZ2}), that is sometimes discontinuous. We also provide examples of `strange' continuous foliations that cannot be straightened by a symplectic homeomorphism.