1. Actions of symplectic homeomorphisms/diffeomorphisms on foliations by curves in dimension 2
- Author
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Arnaud, Marie-Claude and Zavidovique, Maxime
- Subjects
Mathematics::Dynamical Systems ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Computer Science::Programming Languages ,Dynamical Systems (math.DS) ,Mathematics - Dynamical Systems ,Mathematics::Symplectic Geometry - 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., Comment: arXiv admin note: substantial text overlap with arXiv:1809.02372
- Published
- 2022