Actions of symplectic homeomorphisms/diffeomorphisms on foliations by curves in dimension 2

TL;DR

This paper investigates the regularity and structure of invariant foliations under symplectic homeomorphisms and diffeomorphisms in 2D, establishing new regularity results and characterizations of straightenable foliations, with implications for integrable systems.

Contribution

It proves the C1 regularity of generating functions and Hölder continuity of invariant foliations, characterizes straightenable foliations, and shows that Lipschitz integrable twist maps admit Arnol'd-Liouville coordinates.

Findings

Generating function of invariant foliation is C1.

Invariant foliation is Hölder with exponent 1/2.

Lipschitz integrable twist maps have Arnol'd-Liouville coordinates.

Abstract

The two main results of this paper concern the regularity of the invariant foliation of a C0-integrable symplectic twist diffeomorphisms of the 2-dimensional annulus, namely that $\bullet$ the generating function of such a foliation is C1 ; $\bullet$ the foliation is H{\"o}lder with exponent 1/2. We also characterize foliations by graphs that are straightenable via a symplectic homeomorphism and prove that every symplectic homeomorphism that leaves invariant all the leaves of a straightenable foliation has Arnol'd-Liouville coordinates, in which the Dynamics restricted to the leaves is conjugated to a rotation. We deduce that every Lipschitz integrable symplectic twist diffeomorphisms of the 2-dimensional annulus has Arnol'd-Liouville coordinates and then provide examples of 'strange' Lipschitz foliations in smooth curves that cannot be straightened by a symplectic homeomorphism and cannot be invariant by a symplectic twist diffeomorphism.This article is a part of another preprint of the authors, entitled On the transversal dependence of weak K.A.M. solutions for symplectic twist maps, after rewriting ant adding of the H{\"o}lder part.