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Preprints, Working Papers, ... Year : 2018

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

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.

Dates and versions

hal-01871436 , version 1 (10-09-2018)

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Cite

Marie-Claude Arnaud, Maxime Zavidovique. On the transversal dependence of weak K.A.M. solutions for symplectic twist maps. 2018. ⟨hal-01871436⟩
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